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World News | The Latest: Canada Sends More Troops to Eastern Europe
Ottawa, Feb 23 (AP) Prime Minister Justin Trudeau says Canada is sending hundreds more troops to eastern Europe and imposing new sanctions on Russia in response to the deployment of forces into Ukraine.
The prime minister says up to 460 additional Canadian Armed Forces members are being sent to Latvia and the surrounding region to bolster NATO in the face of Russian aggression.
Also Read | Russia-Ukraine Conflict: 27 European Union Nations Unanimously Approve Russian Sanctions.
He also says Canada is taking a number of steps alongside its allies to isolate Russia financially.
Also Read | Russia-Ukraine Crisis: President Vladimir Putin Gets OK to Use Military Force Outside Russia.
WASHINGTON — Secretary of State Antony Blinken says he has cancelled plans to meet his Russian counterpart in Geneva later this week as Russia presses ahead with recognition of separatist regions of Ukraine.
Blinken told reporters on Tuesday that Russia's actions indicated Moscow was not serious about a diplomatic path to resolving the crisis. As a result, he said he had called off his Thursday meeting with Foreign Minister Sergey Lavrov.
Echoing President Joe Biden, Blinken said Russian President Vladimir Putin's decision to recognize the independence of Ukraine's Donbass region was a violation of international law. And, he said the placement of Russian troops there constituted the beginning of an invasion.
Although he held out hope for a peaceful resolution through diplomacy, he said he did not believe a meeting with Lavrov would be productive at this time.
KYIV — Ukraine's president has called up some of the country's military reservists as the threat of a Russian invasion grows, but says there is no need for a full military mobilization.
President Volodymyr Zelenskyy said in a video address to the nation Tuesday night that he had signed a decree to that effect.
"Today there is no need for a full mobilization. We need to quickly add additional staff to the Ukrainian army and other military formations," Zelenskyy said.
He said the decree only applies to those assigned to the so-called operational reserve, which is typically activated during ongoing hostilities, and covers "a special period of time," without clarifying what that means.
"Ukrainians are a peaceful nation, we want silence, but if we keep silent today, we will disappear tomorrow," Zelenskyy said.
There are about 250,000 troops in Ukraine's armed forces.
NEW YORK — Stocks are closing lower on Wall Street after Russia sent forces into Ukraine's eastern regions, escalating tensions.
The benchmark S&P 500 index fell 1% to 4,304.76 on Tuesday, and is now more than 10% below it's all-time high set in January. The Dow Jones Industrial Average and Nasdaq also lost more than 1%. Russian President Vladimir Putin recognized the independence of rebel-held regions of Ukraine, raising fears of an imminent full-scale invasion.
The U.S. and European Union responded with sanctions. Technology shares also weighed on the broader market. Bond yields rose. The yield on the 10-year Treasury rose to 1.93%.
WASHINGTON — President Joe Biden says the U.S. will be sanctioning Russian oligarchs and their families, as well as Russian sovereign debt in retaliation for the country's invasion of Ukraine.
Speaking at the White House on Tuesday, Biden said those sanctions were just the "first tranche" of what the U.S. and its allies stand ready to put in place if Russia launches a larger invasion of Ukraine.
"He's setting up a rationale to take more territory by force," Biden said of recent comments by Russian President Vladimir Putin. "This is the beginning of a Russian invasion of Ukraine."
Biden added that he was authorizing the movement of some U.S. troops in Europe to NATO's Baltic allies as a show of support and solidarity amid the Russian threat.
BERLIN — Germany's Foreign Minister Annalena Baerbock has chaired a call of the G-7 nations in which ministers strongly condemned Russia's recognition of the separatist-controlled areas of Donetsk and Luhansk in eastern Ukraine and the decision to deploy Russian troops there.
Those involved included the foreign ministers of Canada, France, Germany, Italy, Japan, the U.K. and the U.S., along with the European Union.
The German foreign ministry said Tuesday night that the G-7 foreign ministers agreed to step up restrictive measures responding to Russia's actions and reiterated their unwavering commitment to the sovereignty and territorial integrity of Ukraine within its internationally recognized borders.
BERLIN — German Chancellor Olaf Scholz says Germany continues to rule out the delivery of arms to Ukraine despite Russian President Vladimir Putin's order to deploy troops to separatist regions of eastern Ukraine.
Scholz told German public broadcaster ARD on Tuesday night that Germany had made a decision to not export weapons to crisis regions a long time ago, "and we stick to that."
Scholz also pointed out that "Ukraine has a lot of weapons … and the point is that we have to protect Ukraine not by giving more weapons, but by standing together as an international community and saying that we will not simply accept such a breach of international law." He stressed that Germany has been the biggest financial supporter of Ukraine since 2014.
The chancellor added "we must insist that the peace order in Europe is again based on the fact that borders are not moved and that the state sovereignty of countries is not questioned."
BERLIN — Protesters have flocked to the Russian embassy in Berlin to decry Russian President Vladimir Putin's order to deploy troops to separatist regions of eastern Ukraine.
People were waving yellow-blue Ukrainian flags and chanting "We stand united with Ukraine!" as they assembled in front of the building near the German capital's landmark Brandenburg Gate on Tuesday evening.
Some held up banners saying "Ukraine will resist," "Say no to Putin" or "Implement sanctions immediately," while others wrapped themselves into huge Ukrainian flags.
"I'm here to protest against Russia invading our independent Ukrainian territory," said Victoria Baron, 27, who moved from Odesa in Ukraine to Berlin last summer to work for a data science company.
"It's very important that we support our people even though we're abroad," she said adding that she talked to her family back home almost every day and spends hours on social media following the latest developments.
CAIRO — The interim government in chaos-stricken Libya says it rejects Russia's decision to recognize two separatist regions in southeast Ukraine.
In a statement Tuesday, the Government of National Unity has urged Russia to de-escalated and resort to diplomacy to find a solution to the Ukraine crisis. It also called for Russia to withdraw Wagner mercenaries from Libya that fought along with forces of military commander Khalifa Hifter when he unsuccessfully attempted in 2019 to capture the capital of Tripoli..
MOSCOW — The Russian Foreign Ministry said that it has decided to evacuate Russian diplomatic personnel from Ukraine, pointing at threats.
The ministry said Tuesday that Russian diplomats in Ukraine have received multiple threats, adding that they will be evacuated "in the nearest time." It did not elaborate.
The move follows Russia's recognition of Ukraine's rebel regions and the Russian parliament's vote to grant President Vladimir Putin a permission to use military force in Ukraine.
The Ukrainian capital of Kyiv is only a three hour drive from the border of Belarus, where Russia has stationed troops for earlier war games.
BRUSSELS — French foreign minister Jean-Yves Le Drian says the European Union's 27 members have unanimously agreed on an initial set of sanctions targeting Russian officials involved in Ukraine.
EU foreign affairs chief Josep Borrell said Tuesday this first package of sanctions "will hurt Russia and it will hurt a lot."
He said the sanctions would affect members of Russia's duma who voted against the territorial integrity and sovereignty of Ukraine as well as another 27 people and "entities" which are often companies, banks or agencies.
European Commission President Ursula von der Leyen said sanctions will directly target individuals and companies, as well as banks financing "the Russian military apparatus" and contributing to the destabilization of Ukraine.
In addition, von der Leyen said the EU will limit the Russian government's ability to raise capital on the bloc's financial markets.
"We will make it as difficult as possible for the Kremlin to pursue its aggressive actions," she said.
BERLIN — Germany's top security official says authorities are preparing for possible cyberattacks in response to the crisis with Russia.
The Interior Ministry said Tuesday that security agencies have taken protective measures to avert such attacks and Germany's national cyber defense center is monitoring developments.
Interior Minister Nancy Faser said on Twitter that authorities "are prepared for all conceivable impacts of this conflict."
BUCHAREST, Romania — Romanian President Klaus Iohannis said Tuesday after a phone call with his Polish counterpart that they have agreed to hold an "extraordinary summit" Friday in Warsaw between the Bucharest Nine members to "coordinate our response and demonstrate our unity" amid Russia's moves against Ukraine.
"Today I discussed with the President of Poland, Andrzej Duda, the serious security situation in the Black Sea region, following Russian actions, which flagrantly violates international law," Iohannis wrote online. "We stand with Ukraine!"
The Bucharest Nine, which Romania and Poland launched in 2015, is a group comprised of NATO's easternmost members and also includes Hungary, Bulgaria, the Czech Republic, Slovakia, Estonia, Latvia, and Lithuania.
BELVOIR CASTLE, England — Defense ministers from the Baltic states urged world leaders to move swiftly and impose harsh sanctions on Russia, saying their countries know firsthand the dangers of trying to appease a bully.
The Baltic countries' position on NATO's eastern flank was at the forefront of discussions Tuesday as defense ministers from the Joint Expeditionary Force met in central England. The U.K.-led force is a group of 10 nations designed to react more quickly in the event of threats like those now posed by Russia.
Latvian Defense Minister Artis Pabriks said in an interview with the Associated Press that world leaders must act swiftly to impose punishing sanctions or it will be too late to stand up for freedom and democracy.
He said failure to stop the Russian president's "aggression" now would send the message that Moscow can "play around with the Europeans" and the U.S.
LONDON — Sanctions experts say Western measures against Russia will have to go much further to have a chance of deterring President Vladimir Putin from further military intervention in Ukraine.
Tyler Kustra, a University of Nottingham politics professor, said sanctions announced Tuesday by Britain on five Russian banks and three wealthy individuals were "a paper cut."
He said "the U.K. could be doing a lot more. There are certainly far more oil oligarchs in London that could be sanctioned. They could have their assets frozen, they could be kicked out of the country."
Thomas Mayne, a corruption expert and visiting fellow at the Chatham House think-tank, also said Britain's status as a hub for no-questions-asked Russian money was a major problem.
"For at least 20 years, we've been welcoming Russian money," he said. "We've been allowing people with very dubious sources of wealth to gain Tier 1 investor visas in the U.K., to buy property, to list their companies on the London Stock Exchange, far more so than American stock exchanges. And we've created a situation where now we're wondering whether that was such a good idea."
MOSCOW — Russian President Vladimir Putin has called for international recognition of Crimea as part of Russia, an end to Ukraine's NATO membership bid and a halt to weapons shipments there.
Putin claimed Tuesday that Russia's 2014 annexation of Ukraine's Crimean Peninsula should be internationally recognized as a legitimate reflection of the local population's choice, likening it to a vote for Kosovo independence.
The annexation has been widely condemned by Western powers as a breach of international law.
To end the current crisis, he also called for the renunciation of Ukraine's NATO bid, saying it should assume a "neutral status," and said that the West should stop sending weapons there.
BRUSSELS — NATO Secretary-General Jens Stoltenberg says Russia is taking military action against Ukraine and has condemned Moscow's decision to recognize separatist areas of southeast Ukraine as independent.
Stoltenberg called the move Tuesday a "serious escalation by Russia and a flagrant violation of international law." The NATO chief called the military action a "further invasion" of Ukraine by Russia which had already invaded its neighbor in 2014.
He added that there's "every indication" Russia continues to plan for a full-scale attack on Ukraine.
Stoltenberg said that NATO allies have more than 100 warplanes on high alert and more than 120 warships ready at sea from the Arctic Circle to the Mediterranean Sea.
He said that the NATO response force remains on high readiness but is not yet being deployed, although some allies are moving troops, ships and planes into the Baltic states and near the Black Sea to defend other NATO members.
MOSCOW — Russian lawmakers have given President Vladimir Putin permission to use military force outside the country.
The unanimous vote in Russia's upper house on Tuesday could presage a broader attack on Ukraine after the U.S. said an invasion was already underway there.
The vote formalizes a Russian military deployment to the rebel regions, where an eight-year conflict has killed nearly 14,000 people.
MOSCOW — Russia's closest allies appeared reluctant to immediately back Moscow's decision to recognize the independence of rebel-held areas in eastern Ukraine.
The Foreign Ministry of Belarus said Tuesday it saw Russian President Vladimir Putin's move "with respect and understanding," but refrained from saying whether Minsk would follow suit and recognize the self-proclaimed Donetsk and Luhansk republics.
Officials in Kazakhstan, whose president last month asked for a Russia-led security alliance to send troops to quell violent unrest, said the issue of recognizing the separatist regions was not on the country's agenda.
Azerbaijani President Ilham Aliyev, who's in Moscow for talks with Putin, made no mention of Moscow's recognition of Donetsk and Luhansk in his publicly broadcast remarks, but focused instead on bilateral relations.
Another Russian ally, Armenia, has so far issued no message of support for the Russian move.
Meanwhile, Putin sought to reassure Russia's ex-Soviet allies that he doesn't envision resurrecting the Soviet Union. He told the Azerbaijani president that speculation that Russia would attempt to restore its empire "is absolutely not true."
WASHINGTON — The White House has begun referring to Russian troop deployments in eastern Ukraine as an "invasion" after initially hesitating to use the term — a red line that President Joe Biden has said would result in the U.S. levying severe sanctions against Moscow.
Jon Finer, the principal deputy national security adviser, said Tuesday: "We think this is, yes, the beginning of an invasion, Russia's latest invasion into Ukraine." He said "latest" was important — "an invasion is an invasion and that is what is under way."
The White House decided to begin referring to Russia's actions as an "invasion" because of the situation on the ground, according to a U.S. official who spoke on the condition of anonymity to discuss internal deliberations.
The administration resisted initially calling the deployment of troops because the White House wanted to see what Russia was actually going to do. The official added that, after assessing Russian troop movements, it became clear it was a new invasion.
— By Aamer Madhani in Washington.
LONDON — Prime Minister Boris Johnson says Britain is slapping sanctions on five Russian banks and three wealthy individuals over Russia's latest military moves on Ukraine.
Johnson told lawmakers that sanctions would hit Rossiya Bank, IS Bank, General Bank, Promsvyazbank and the Black Sea Bank. He said three Russian oligarchs with interests in energy and infrastructure — Gennady Timchenko, Boris Rotenberg and Igor Rotenberg — will have their assets frozen and be banned from traveling to the U.K.
All three have already been sanctioned by the United States.
Johnson accused Russian President Vladimir Putin of "establishing the pretext for a full-scale offensive" against Ukraine and said "further powerful sanctions" would follow if that happened.
"This the first tranche, the first barrage of what we are prepared to do and we hold further sanctions at readiness to be deployed," Johnson told British lawmakers.
He also said Russian tanks and armored personnel carriers have been spotted in the separatist Ukrainian regions recognized as independent by Putin. He said that amounts to "a renewed invasion" of Ukraine.
BERLIN — Chancellor Olaf Scholz says Germany has taken steps to halt the process of certifying the Nord Stream 2 gas pipeline from Russia.
Scholz told reporters in Berlin on Tuesday that his government was taking the measure in response to Moscow's actions in Ukraine.
The pipeline bringing natural gas from Russia to Germany has long been criticized by the United States and some European countries who argue that it increases Europe's reliance on Russian energy supplies.
Scholz said that the government had decided to "reassess" the certification of the pipeline, which hasn't begun operating yet.
"That will certainly take time, if I may say so," he said.
MOSCOW — Russia says its recognition of independence for areas in eastern Ukraine extends to territory currently held by Ukrainian forces.
The statement Tuesday further raises the stakes amid Western fears that Moscow could follow up on Monday's recognition of rebel regions with a full-fledged invasion of Ukraine.
Kremlin spokesman Dmitry Peskov said that Russia has recognized the rebel regions' independence "in borders that existed when they proclaimed" their independence in 2014.
Ukrainian forces later reclaimed control of large part of both regions during a nearly eight-year conflict that has killed over 14,000 people. (AP)
World News | 4 UN Security Officers Detained in Central African Republic
Trump Wants Putin To Send Russian Troops To The US To Protect The Southern Border


text:
\section{Introduction}

The reconstruction of conductivity distributions in two or three
dimensions from measurements of the current density-to-voltage map is known as
electrical impedance tomography, or EIT, and has applications in
medical imaging, nondestructive testing, and geophysics. For the 3-D
bounded domain considered here, medical applications include head
imaging and the detection of breast tumors.  See, for example,
\cite{holder_book} for a survey of clinical applications of EIT. In
this work, we consider a bounded domain in $\mathbf R^3$ and present a direct reconstruction algorithm and its
numerical implementation on the unit sphere. The theoretical foundation of the method dates back
more than 20 years to a series of papers by
Sylvester-Uhlmann \cite{sylvesterUhlmann87}, Novikov \cite{novikov},
Nachman-Sylvester-Uhlmann \cite{NSU} and Nachman \cite{nachman88}.
The algorithm makes use of complex geometrical optics (CGO) solutions
to the Schr\"odinger equation and uses the inverse scattering method.  This is described in detail
in section 2 of this paper.

The inverse conductivity problem was first formulated mathematically
by Calder\'on \cite{Calderon} as follows. Let $\Omega \subset \mathbf R^n, n
\geq 3$ be a simply connected, bounded domain with smooth boundary
$\partial \Omega,$ and let $\gamma \in L^\infty(\Omega)$ denote the
conductivity distribution. Assume there exists $C>0$ such that for $x\in\Omega$,
  $C^{-1} \leq \gamma(x) \leq C$.
The electric potential $u$ arising from the application of a known voltage to the boundary of $\Omega$ is
modeled by the generalized Laplace equation with Dirichlet boundary condition
\begin{eqnarray}  \label{eq:cond}
  \nabla \cdot \gamma \nabla u = 0 \mbox{ in } \Omega,\qquad u = f  \mbox{ on } \partial \Omega.
\end{eqnarray}
The Dirichlet-to-Neumann map $\Lambda_\gamma$
is defined by
\begin{eqnarray}  \label{DNmap}
  \Lambda_\gamma f = \gamma \frac{\partial u}{\partial \nu}\vert_{\partial \Omega}.
\end{eqnarray}
Thus $\Lambda_\gamma$ represents static electrical boundary
measurements: it maps an applied voltage distribution on the boundary
to the resulting current flux through the boundary. Calder\'on
\cite{Calderon} posed the question of whether the conductivity
$\gamma$ is uniquely determined by the Dirichlet-to-Neumann map, and
if so, how to reconstruct the conductivity. He gave an affirmative
answer to the uniqueness question for the linearized problem and gave
a reconstruction algorithm for that case.  His algorithm is described
in section~\ref{sec:calderon} of this paper.

The uniqueness question for $\gamma\in L^{\infty}(\Omega)$ is still open
in $\mathbf R^3$, but has been solved recently by Astala and P\"aiv\"arinta
\cite{astalaPaivarinta} for a bounded domain in $\mathbf R^2$, sharpening the previous results due to Nachman
\cite{nachman96} in which $\gamma\in W^{2,p}(\Omega)$, $p>1,$ and Brown
and Uhlmann \cite{brownUhlmann} in which $\gamma\in W^{1,p}(\Omega)$, $p>2$.
In three dimensions the uniqueness problem was solved for smooth conductivities
in \cite{sylvesterUhlmann87}. At the time of
this publication, in $\mathbf R^3$ the uniqueness results with lowest
regularity are \cite{brownTorres} with $\gamma\in W^{3/2,p}(\Omega)$, $p>2n$
and \cite{PPU} with $\gamma\in W^{3/2,\infty}(\Omega)$.

Most existing 3-D EIT reconstruction algorithms are linear or
iterative, minimizing a functional that describes the
nearness of the predicted voltages to the measured data in a given
norm with one or more regularization terms.  In contrast, the algorithm presented here is direct and fully nonlinear.
It is similar to the 2-D D-bar algorthims based on the works \cite{nachman96} and \cite{brownUhlmann}, which were first
implemented in \cite{SMI00, ams, knudsen03,MS03}.  In these initial works the Born approximation to the CGO solutions is used in the
computation of the scattering transform.  It was used successfully on experimental tank data in, for
example, \cite{isaacson_reconstruct_chest_2004,murphy_2009} and human chest data in \cite{isaacson_imaging_cardiac_2006,DeAngeloMueller}.  This
inspired the approach in section \ref{sec:texp} of this paper in which the Born approximation is used in the 3-D direct algorithm.  For further
reading on 2-D D-bar algorithms, the reader is referred to
\cite{knudsenLassasMuellerSiltanen07} in which the application to discontinuous conductivity distributions is specifically addressed,
and \cite{knudsenLassasMuellerSiltanen09} in which a rigorous regularization
framework is established using the full scattering transform.
Calder\'on's method has
also recently been used for the reconstruction from experimental data
in both 2-D \cite{bikowskiMueller} and 3-D  \cite{Greg_Dave}.

In this work, we assume the conductivity $\gamma\in C^2(\overline \Omega),$ we
take $\Omega$ to be the unit sphere in $\mathbf R^3,$ and assume $\gamma = 1$
near $\partial\Omega.$ The smoothness assumption on $\gamma$ is necessary, but
the other assumptions are made mainly for simplicity in the numerical
computations. We stress in particular that the theory is valid in more
complex geometries.  The study of the effects of noise in the data is not in the scope of this
paper, but rather is left for future work.

The outline of the paper is as follows. In section 2.2 we describe a
direct reconstruction algorithm with a linearizing assumption
tantamount to a Born approximation. That approach is referred to as
the $\mathbf{t}^{\mbox{{\tiny exp}}}$ approach, consistent with the notation used in the 2-D
D-bar algorithms.  An explicit connection to the linearized method of
Calder\'on is established in section \ref{sec:calderon}.  The
reconstruction of the conductivity in the 2-D D-bar method described
in the works above is achieved by taking a small frequency limit in a
D-bar equation for the CGO solutions to directly obtain $\gamma(x)$.
In contrast, here we have to take a high complex frequency limit.  A
D-bar equation for the 3-D problem is utilized in
\cite{corneanKnudsenSiltanen}, resulting in a promising, but more
complicated approach than the one studied here.  The numerical
implementation of that approach is left for future work.  In section
\ref{sec:spherical} we consider the case of spherically symmetric
conductivities and show symmetry properties in the scattering
transform. We also show how the Dirichlet-to-Neumann map can be
represented and approximated in that case. Details on the numerical
implementation are found in section \ref{sec:numerics}.  Numerical
examples are found in section \ref{sec:examples}.

\section{The reconstruction methods} \label{sec:recon} In this section
we describe the theoretical reconstruction method, the $\mathbf{t}^{\mbox{{\tiny exp}}}$ approach, and
the relationship to Calder\'on's linearized method.

\subsection{The nonlinear reconstruction method}\label{sec:nonlinear}
The method was developed in
\cite{sylvesterUhlmann87,novikov,NSU,nachman88}; here we provide a
brief outline.  The reader is referred to \cite{nachman88} for
rigorous proofs.  The equations closely parallel those of the 2-D
problem (note that \cite{nachman88} precedes that work), and so
readers familiar with that case will recognize the notation and
functions involved.

The initial step is to transform the conductivity equation into a
Schr\"odinger equation. Indeed, if $u$ satisfies \eqref{eq:cond} then
$v= \gamma^{1/2}u$ satisfies
\begin{eqnarray}\label{eq:schr}
  (-\Delta + q) v = 0 \textnormal { in } \Omega ~~~ \mbox{with} ~~~  q = \frac {\Delta \gamma^{1/2}}{\gamma^{1/2}}.
\end{eqnarray}
Note that $q = 0$ near $\partial\Omega.$ The Dirichlet-to-Neumann map
for equation \eqref{eq:schr} is defined by
 $$ \Lambda_q f =  \frac{\partial v}{\partial \nu}|_{\partial \Omega},$$
where now $v$ satisfies \eqref{eq:schr} with $v|_{\partial\Omega} = f.$
In general the maps $\Lambda_\gamma$ and $\Lambda_q$ are  related by
\begin{eqnarray} \label{DtNmaps}
\Lambda_q = \gamma^{-1/2}\left(\Lambda_{\gamma}+\frac{1}{2}\frac{\partial\gamma}{\partial\nu}\right)\gamma^{-1/2}.
\end{eqnarray}
The assumption that $\gamma = 1$ in a neighborhood of $\partial \Omega$ simplifies \eqref{DtNmaps}
to $\Lambda_q = \Lambda_\gamma.$

To define the CGO solutions, introduce a complex frequency parameter
$\zeta \in {\mathbb C}^3$ and define the set
\begin{eqnarray} \label{vdef}
\V = \{{\mathbb C}^3\setminus\{0\} \colon \zeta\cdot\zeta =0\}.
\end{eqnarray}
Then $e^{ix\cdot\zeta}$ is
 harmonic in $\mathbf R^3$ if and only if
$\zeta \in \V.$ For $\xi\in\mathbf R^3$, introduce the subset of $\V$ given by
\begin{eqnarray}
  \label{eq:Vxi}
   \V_\xi = \{ \zeta \in \V \colon (\xi + \zeta)^2 = 0\}.
\end{eqnarray}
Note that $\zeta\cdot \zeta = (\xi + \zeta)^2 = 0$ gives an explicit
characterization of $\V_\xi$ in terms of an auxiliary vector
$\xi^\perp \in \mathbf R^3$ with $ \xi^\perp\cdot \xi = 0.$ Indeed suppose
$\zeta_R, \zeta_I \in \mathbf R^3.$ Then $\zeta = \zeta_R + i
\zeta_I\in\V_\xi$ if and only if
\begin{eqnarray}\label{zetaform}
     \zeta_R &= -\xi/2 +
    \xi^\perp,\\
    \zeta_I \cdot \xi = \zeta_I \cdot \xi^\perp &= 0 ,\quad |\zeta_I| = |\zeta_R|.\nonumber
\end{eqnarray}

Since $q=0$ in a neighborhood of $\partial\Omega$, one can extend $q=0$ into $\mathbf R^3\setminus \overline\Omega.$ The CGO solutions
$\psi(x,\zeta)$ to the Schr\"odinger equation solve
\begin{eqnarray}
  \label{schrodinger}
  (-\Delta + q(x))\psi(x,\zeta) = 0, \quad x\in \mathbf R^3, \quad \zeta \in \V,
\end{eqnarray}
and behave like $e^{ix\cdot\zeta}$ for $|\zeta|$ large. More precisely,
define
$$\mu(x,\zeta) = \psi(x,\zeta) e^{-ix\cdot\zeta}.$$
 Then $\mu-1$ approaches
zero in  a certain sense as either $|x|$ or $|\zeta|$ tends to infinity, see \cite{sylvesterUhlmann87,nachman88}.  Note that
$\psi(x,\zeta)$ grows exponentially for $x\cdot\Im \zeta <0$. The function $\mu$ satisfies
\begin{eqnarray}\label{mueq}
  (-\Delta -2i\zeta\cdot\nabla +  q) \mu(x,\zeta) = 0 \textnormal { in } \mathbf R^3.
\end{eqnarray}

Denote by $G_\zeta$ the Faddeev Green's function defined by
\begin{eqnarray}\label{Gzeta}
  G_\zeta (x) = e^{ix\cdot \zeta}g_\zeta(x), ~~~ \mbox{where} ~~~ g_\zeta(x) = \frac{1}{(2\pi)^3}\int_{\mathbf R^3} \frac{e^{ix\cdot
  \xi}}{|\xi|^2 + 2\xi\cdot \zeta}d\xi,
\end{eqnarray}
 where the integral is understood in the sense of the Fourier transform defined on the space of tempered
distributions. The functions $G_\zeta,g_\zeta$ are fundamental
solutions of the Laplace equation
and conjugate Laplace equation respectively, i.e.\
\begin{eqnarray}
  \Delta_\zeta g_\zeta = -\delta_0 \quad \mbox{and} \quad  \Delta G_\zeta = -\delta_0.
\end{eqnarray}
Then \eqref{mueq} is equivalent to the Faddeev-Lippmann-Schwinger equation
\begin{eqnarray}\label{muLS}
  (I + g_\zeta \ast (q \;\cdot\;))\mu = 1 \textnormal { in } \mathbf R^3.
\end{eqnarray}
Estimates for the operator $g_\zeta \ast$ for large $\zeta$
(\cite{sylvesterUhlmann87}) and small $\zeta$
(\cite{corneanKnudsenSiltanen}) give the existence and uniqueness of
$\mu$ (and therefore $\psi$) for any sufficiently large or small $\zeta \in
\V.$

The key intermediate object in the reconstruction method is the
so-called non-physical scattering transform of the potential $q$
defined for $\xi\in\mathbf R^3$ and sufficiently large or small $\zeta \in
\V$ by
\begin{eqnarray}
  \t(\xi,\zeta) = \int_{\Omega}
  e^{-ix\cdot(\xi+\zeta)}\psi(x,\zeta)q(x)dx. \label{t}
\end{eqnarray}
Integrating by parts and assuming that $\zeta \in \V_ \xi$
we find that
\begin{eqnarray}\label{tboundary}
  \t(\xi,\zeta) =
  \int_{\partial\Omega}e^{-ix\cdot(\xi+\zeta)}(\Lambda_q-\Lambda_0)\psi(x,\zeta)d\sigma(x).
\end{eqnarray}
Thus we require $\psi|_{\partial\Omega}$ in order to compute the
scattering transform from the Dirichlet-to-Neumann map. It turns out
that
$\psi|_{\partial\Omega}$ satisfies a uniquely solvable Fredholm
integral equation of the second kind  on $\partial\Omega$ \cite{novikov,nachman88},  namely,
\begin{eqnarray}
  \label{eq:bie}
  \psi(x,\zeta)+ \int_{\partial\Omega} G_{\zeta}(x-\tilde{x})
  (\Lambda_q-\Lambda_0)\psi(\tilde{x},\zeta) d\sigma(\tilde{x})= e^{ix\cdot
    \zeta}, \quad x\in\partial\Omega.
\end{eqnarray}

Note from \eqref{t} and the fact that $\psi\sim e^{i\zeta\cdot x}$ that from the scattering transform one can compute the Fourier transform $\hat q$ of
the potential by taking the large frequency limit
\begin{eqnarray}
  \label{eq:zetalim}
  \lim_{|\zeta| \rightarrow \infty}\t(\xi,\zeta) = \hat q (\xi).
\end{eqnarray}

Summary of the reconstruction method:
\begin{enumerate}
\item\label{nonlinearStep1} Solve the boundary integral equation  \eqref{eq:bie} for
  $\psi|_{\partial\Omega}$.
  \item  Compute $\t(\xi,\zeta)$ for $\xi \in \mathbf R^3,  \; \zeta \in \V_\xi$ by \eqref{tboundary}.
\item Compute $\hat q (\xi)$ from \eqref{eq:zetalim} .
\item Compute $q$ by inverting the Fourier transform.
\item Compute $\gamma$ by solving $-\Delta\sqrt{\gamma}+q\sqrt{\gamma}=0$ in $\Omega,$ $\sqrt{\gamma}|_{\partial\Omega}=1.$
\end{enumerate}

We stress that the ill-posedness of the inverse problem is in this
algorithm isolated in the first step.

\subsection{The reconstruction method using $\mathbf{t}^{\mbox{{\tiny exp}}}$}\label{sec:texp}

Inspired by the $\mathbf{t}^{\mbox{{\tiny exp}}}$ approximation in the 2-D D-bar method, an analogous approach can be taken in 3-D.
Approximating $\psi(x,\zeta)$ on the boundary by its asymptotic
behavior  $e^{ix\cdot \zeta}$ eliminates the need for the ill-posed
first step. We define for $\xi\in\mathbf R^3, \zeta\in \V_\xi$
\begin{eqnarray}\label{Texp}
\mathbf{t}^{\mbox{{\tiny exp}}}(\xi,\zeta)  =  \int_{\partial\Omega}e^{-ix\cdot(\xi+\zeta)}(\Lambda_q-\Lambda_0) e^{ix\cdot \zeta}d\sigma(x).
\end{eqnarray}
This approximation is tantamount to a linearization of the first step in the reconstruction
algorithm above around $\gamma=1$.  Using $\mathbf{t}^{\mbox{{\tiny exp}}}$ for $\t$ in \eqref{eq:zetalim} gives the following simple reconstruction
algorithm:
\begin{enumerate}
\item Compute $\mathbf{t}^{\mbox{{\tiny exp}}}(\xi,\zeta)$ for $\xi \in \mathbf R^3, \; \zeta \in
  \V_\xi$ by \eqref{Texp}.
\item Compute
  \begin{eqnarray}\label{texpzetalim}
    \widehat q^{\mbox{{\tiny exp}}} (\xi) = \lim_{|\zeta|\rightarrow
      \infty}\mathbf{t}^{\mbox{{\tiny exp}}}(\xi,\zeta)
  \end{eqnarray}
  and then $q^{\mbox{{\tiny exp}}}$ by inverting the Fourier transform.
\item Compute $\mathbf{\gamma}^{\mbox{{\tiny exp}}}$ by solving $-\Delta\sqrt{\mathbf{\gamma}^{\mbox{{\tiny exp}}}} +q^{\mbox{{\tiny exp}}}
  \sqrt{\mathbf{\gamma}^{\mbox{{\tiny exp}}}}=0$ in $\Omega$  $\sqrt{\mathbf{\gamma}^{\mbox{{\tiny exp}}}}|_{\partial\Omega}=1.$
\end{enumerate}
It is not guarenteed from the theory that the limit in \eqref{texpzetalim} is
well-defined. In our numerical simulations we will compute
$\mathbf{t}^{\mbox{{\tiny exp}}}(\xi,\zeta)$ for a fixed but large value of $\zeta.$ This will
numerically define $\widehat q^{\mbox{{\tiny exp}}} (\xi).$

\subsection{Calderon's linearized reconstruction method} \label{sec:calderon}

Several properties of $\mathbf{t}^{\mbox{{\tiny exp}}}$ can be
established from an analysis comparing this approach to that of
Calder\'on.  In \cite{knudsenLassasMuellerSiltanen07} a connection was established between the 2-D
D-bar method based on the global uniqueness proof by Nachman \cite{nachman96}
and Calder\'on's linearized reconstruction method.

Define a function ${u}^{\mbox{{\tiny exp}}}(x,\zeta)$ as the unique solution to the boundary value problem
\begin{eqnarray*}
\nabla\cdot\gamma\nabla{u}^{\mbox{{\tiny exp}}}(x,\zeta) &= 0, \quad x\in \Omega, \quad \zeta\in {\mathbb C}^3 \\
{u}^{\mbox{{\tiny exp}}}\vert_{\partial\Omega} &= e^{ix\cdot\zeta}.
\end{eqnarray*}
Integration by parts in equation \eqref{Texp} results in a formula for $\mathbf{t}^{\mbox{{\tiny exp}}}$
defined in terms of $\gamma$ in the interior
\begin{eqnarray} \label{texp_int}
\mathbf{t}^{\mbox{{\tiny exp}}}(\xi,\zeta) &=& \int_{\Omega}(\gamma - 1)\nabla{u}^{\mbox{{\tiny exp}}}(x,\zeta)\cdot\nabla e^{-ix\cdot(\xi+\zeta)}dx.
\end{eqnarray}
Write ${u}^{\mbox{{\tiny exp}}} = e^{ix\cdot\zeta} + \delta u$ for $\delta u\in H^1_0(\Omega)$.  Then $\delta u$ satisfies
\begin{eqnarray}
\nabla\cdot(\gamma\nabla\delta u) &=& -\nabla\cdot((\gamma-1)\nabla e^{ix\cdot\zeta}) \label{du_pde},
\end{eqnarray}
and one can estimate
\begin{eqnarray}\label{du_estimate}
\|\delta u\|_{H^1(\Omega)}\leq C\|(\gamma-1)\nabla
e^{ix\cdot\zeta}\|_{L^2(\Omega)} \leq |\zeta|\|\gamma - 1\|_{L^{\infty}(\Omega)}e^{|\zeta|R},
\end{eqnarray}
where $R$ is the radius of the smallest ball containing $\Omega$. From
\eqref{texp_int} we then get
\begin{eqnarray*}
  \mathbf{t}^{\mbox{{\tiny exp}}}(\xi,\zeta) &= \int_{\Omega}(\gamma - 1)\nabla(e^{ix\cdot\zeta}
  + \delta u)\cdot\nabla e^{-ix\cdot(\xi+\zeta)}dx\\
  &= \int_{\Omega}(\gamma - 1)\nabla e^{ix\cdot\zeta}
  \cdot\nabla e^{-ix\cdot(\xi+\zeta)}dx + R(\xi,\zeta)\\
  &= (\xi\cdot \zeta) \int_{\Omega}(\gamma - 1)e^{-ix\cdot\xi}dx +
  R(\xi,\zeta),
\end{eqnarray*}
where the remainder term
\begin{eqnarray*}
   R(\xi,\zeta) = \int_{\Omega}(\gamma - 1)\nabla \delta u \cdot\nabla e^{-ix\cdot(\xi+\zeta)}dx.
\end{eqnarray*}
Since $(\xi + \zeta)^2 = \zeta^2 = 0$ we have $-\xi^2 = 2\xi \cdot
\zeta$ and hence
\begin{eqnarray}\label{FTgamma}
   \mathbf{t}^{\mbox{{\tiny exp}}}(\xi,\zeta) =  -\frac{|\xi|^2}2 \widehat{(\gamma - 1)} (\xi) + R(\xi,\zeta).
\end{eqnarray}
The remainder is $\O (|\zeta|)$ for $\zeta$ small,  which can be seen from
\eqref{du_estimate}. This fact suggests that we use the minimal $\zeta
\in \V_\xi,$ that is
\begin{eqnarray*}
  \zeta_\xi = -\frac \xi 2 + i\zeta_I, \qquad \textnormal {with }  \zeta_I
  \cdot \xi = 0,\; |\zeta_I| = \frac{|\xi|}2.
\end{eqnarray*}
Moreover, with this particular choice we can divide in \eqref{FTgamma}
by $|\xi|^2$  as the following proposition shows.
\begin{prop}
  Suppose $\gamma \in L^\infty(\Omega).$ Then
  $$|\mathbf{t}^{\mbox{{\tiny exp}}}(\xi,\zeta_\xi)| = \O(|\xi|^2)$$ for small $|\xi|.$
\end{prop}
\begin{proof}
  Note that $|\zeta_\xi|^2 =|\xi|^2/2.$ Since $\Lambda_{\gamma}$ maps constant functions to zero and has its range inside
  the space of mean free functions in $H^{-1/2}(\partial\Omega)$, we have that
  for small $|\xi|$
  \begin{eqnarray*}
    |\mathbf{t}^{\mbox{{\tiny exp}}}(\xi,\zeta_\xi)| &=
    \left|\int_{\partial\Omega}(e^{-ix\cdot (\xi +
      \zeta_\xi)}-1)(\Lambda_\gamma-\Lambda_1)(e^{ix\cdot
      \zeta_\xi}-1)d\sigma(x)\right|\\
& \leq C |\xi + \zeta_\xi| |\zeta_\xi|,
  \end{eqnarray*}
and hence the particular form of $\zeta_\xi$ gives the conclusion.
\end{proof}
With the particular choice $\zeta =\zeta_\xi$ in \eqref{FTgamma} we
now neglect the term $R(\xi,\zeta_\xi)$ and divide by $-|\xi|^2,$ which
gives
\begin{eqnarray*}
   -2\frac{\mathbf{t}^{\mbox{{\tiny exp}}}(\xi,\zeta_\xi)}{|\xi|^2} \approx  \widehat{(\gamma - 1)}(\xi).
\end{eqnarray*}
Introduce $\chi_B(\xi),$ the characteristic function on the ball
$|\xi|<B.$ With this function we remove high frequencies  and invert the Fourier transform.
This results in a linear reconstruction algorithm
\begin{eqnarray}\label{gammaapp}
  \mathbf{\gamma}^{\mbox{{\tiny app}}}( x) = 1 -\frac {2}{(2\pi)^3} \int_{\mathbf R^3} \frac{\mathbf{t}^{\mbox{{\tiny exp}}}(\xi,\zeta_\xi)}{|\xi|^2}
   e^{ix\cdot \xi} \chi_B(\xi)d\xi.
\end{eqnarray}
This formula is equivalent to the second inversion formula obtained by Calder\'on \cite[p.\ 72]{Calderon}.

In summary the linear reconstruction algorithm consists of two steps:
\begin{enumerate}
\item Compute $\mathbf{t}^{\mbox{{\tiny exp}}}(\xi,\zeta_\xi)$ by \eqref{Texp}.
\item Compute the reconstruction by \eqref{gammaapp}.
\end{enumerate}

This method is truly a linearization of the nonlinear reconstruction method
outlined in section \ref{sec:nonlinear}. As explained above $\mathbf{t}^{\mbox{{\tiny exp}}}$
is a linearization of the first step on page
\pageref{nonlinearStep1}. Moreover, the computation of the quantity
\begin{eqnarray*}
  1 - \frac {1}{(2\pi)^3} \int_{\mathbf R^3}
  \frac{\mathbf{t}^{\mbox{{\tiny exp}}}(\xi,\zeta_\xi)}{|\xi|^2} e^{ix\cdot \xi} \chi_B(\xi)d\xi
\end{eqnarray*}
linearizes the step $\hat q \mapsto \sqrt\gamma.$ Finally,
linearizing the square function gives \eqref{gammaapp}.

\section{The case of a spherically symmetric conductivity} \label{sec:spherical}

As a test problem it is of special interest to consider spherically
symmetric conductivities in the unit sphere.  In this case  the
scattering transform has certain symmetry properties described below. Moreover, the
Dirichlet-to-Neumann map is described explicitly  in terms of
eigenvalues and eigenfunctions, which in this case are
the spherical harmonics. These properties will be
derived in this section.

\subsection{Symmetry in the scattering transform} \label{sec:symmetries}
The Fourier transform of a spherically symmetric function is
spherically symmetric itself. For the scattering transforms $\t$ and
$\mathbf{t}^{\mbox{{\tiny exp}}}$ we have similar prperties. In the following we will tacitly
assume that $\zeta$ is either small or large such that ${\mathbf{t}}(\xi,\zeta)$ is well-defined.
\begin{prop}
  Let $R\in SO(2)$ be arbitrary, and suppose $q(x) = q(Rx)$ for $x \in \Omega.$ Then
  \begin{eqnarray}\label{Tsymm1}
    \t(\xi,\zeta) = \t(R\xi,R\zeta), \quad \mathbf{t}^{\mbox{{\tiny exp}}}(\xi,\zeta) = \mathbf{t}^{\mbox{{\tiny exp}}} (R\xi,R\zeta)
  \end{eqnarray}
  In particular,
\begin{eqnarray}\label{Tsymm2}
  \t(\xi,\zeta_1) = \t(\xi,\zeta_2), \quad \mathbf{t}^{\mbox{{\tiny exp}}}(\xi,\zeta_1) = \mathbf{t}^{\mbox{{\tiny exp}}}(\xi,\zeta_2)
\end{eqnarray}
for all $\zeta_1,\zeta_2 \in \V_\xi.$
\end{prop}
\begin{proof}
  We will prove the result for $\t$ only; for $\mathbf{t}^{\mbox{{\tiny exp}}}$ the reasoning
  is similar. Let $\mathbf R \in SO(2).$ By the uniqueness of the CGO
  solutions, the rotational invariance of the Laplace operator, and the
  symmetry in $q$ we
  have $\psi(x,\zeta) = \psi (Rx, R\zeta).$ Consider the integral
  \eqref{t} and make the change of variables $R^Ty = x:$
  \begin{eqnarray*}
    \t(\xi,\zeta) &=   \int_{\Omega}
  e^{-ix\cdot(\xi+\zeta)}\psi(x,\zeta)q(x)dx \\
  &= \int_{R\Omega}
  e^{-i R^Ty \cdot (\xi+\zeta)}\psi(R^Ty,\zeta)q(R^Ty)d(R^Ty)\\
  &= \int_{\Omega}
  e^{-i y \cdot R(\xi+\zeta)}\psi(y,R\zeta)q(y)dy\\
  &= \t(R\xi,R\zeta).
  \end{eqnarray*}

To prove \eqref{Tsymm2} fix $\xi \in \mathbf R^3$ and take $\zeta_1,\zeta_2 \in \V_\xi$ with
$|\zeta_1| = |\zeta_2|.$ Then
\begin{eqnarray*}
  \zeta_j = \Re(\zeta_j) + i \Im(\zeta_j),\quad \Re(\zeta_j) & =-\frac
  \xi 2 + \xi_j^\perp \; \xi_j^\perp \cdot \xi = 0,\\
 \Im(\zeta_j) \cdot \xi &= \Im(\zeta_j) \cdot \xi^\perp = 0.
\end{eqnarray*}
Define a linear transformation $R$ by $R\xi = \xi,\; R \xi_1^\perp =
\xi_2^\perp,\; R (\xi \times \xi_1^\perp) = \xi \times \xi_2^\perp.$ As
a consequence $R \in SO(2)$ and \eqref{Tsymm2} follows from
\eqref{Tsymm1}.
\end{proof}

The Fourier transform of a real and even function is real itself. For
the scattering transform we have the following equivalent property:

\begin{prop}
  Suppose $q(x)$ is real and even. Then for $ \xi \in \mathbf R^3, \; \zeta \in \V_\xi$
  \begin{eqnarray}
    \overline{\t(\xi,\zeta)} = \t(\xi,\overline\zeta), \label{Tsymm3}\quad  \overline{\mathbf{t}^{\mbox{{\tiny exp}}}(\xi,\zeta)} = \mathbf{t}^{\mbox{{\tiny exp}}}(\xi,\overline\zeta).
  \end{eqnarray}
\end{prop}
\begin{proof}
We will again only show the properties for $\t.$  From the uniqueness
of the CGO solutions it follows that if $q$ is even then
$\mu(-x,\zeta) = \mu(x,-\zeta).$ Moreover, if $q$ is real, then
$\overline{\mu(x,\zeta)} = \mu(x,-\overline\zeta).$ Hence, if $q$ is both even and
real then
\begin{eqnarray*}
  \overline{\t(\xi,\zeta)} &= \int_\Omega e^{ix\cdot\xi} \overline {q(x)}
  \overline{\mu(x,\zeta)}dx  = \int_\Omega e^{-i(-x)\cdot\xi}  {q(x)}{\mu(x,-\overline\zeta)}dx\\
  & = \int_\Omega e^{-iy\cdot\xi}  {q(y)}
  \overline{\mu(y,\overline\zeta)}dy\\
  &=\t(\xi,\overline\zeta).
\end{eqnarray*}
\end{proof}
We now have a corollary for spherically symmetric
potentials:
\begin{cor}\label{Treal}
  Suppose $q$ is spherically symmetric. Then
  \begin{eqnarray*}
    \overline{\t(\xi,\zeta)} =\t(\xi,\zeta),\quad \xi \in \mathbf R^3, \; \zeta \in \V_\xi.
  \end{eqnarray*}
\end{cor}
\begin{proof}
  There exists $R \in SO(2)$ such that
  \begin{eqnarray*}
    R(\xi) =\xi,\quad R(\zeta) = \overline \zeta,
  \end{eqnarray*}
  and hence from \eqref{Tsymm1} we have $\t(\xi,\zeta)
  = \t(\xi,\overline\zeta).$ Equation \eqref{Tsymm3} now implies the result
  for $\t.$ For $\mathbf{t}^{\mbox{{\tiny exp}}}$ the result follows similarly.
\end{proof}

\subsection{Eigenfunctions and eigenvalues for the Dirichlet-to-Neumann map}

We use the same
ideas for the computation of eigenvalues for the 3-D problem that were used for the 2-D problem in  \cite{SMI00}.

\begin{prop}
Let $D$ be the unit disk and suppose $\gamma(x)$ is  spherically symmetric. Then the eigenfunctions of $\Lambda_\gamma$
are the spherical harmonics $Y_l^m.$
\end{prop}
\begin{proof}
  When $\gamma$ is spherically symmetric it follows from separation of
  variables that the solution to $\nabla \cdot
  \gamma \nabla u_{lm} = 0$ with $u_{lm}|_{\partial D} = Y_l^m$ is
  \begin{eqnarray}\label{eq:sphericalsol}
    u_{lm} = R_{l}(r)Y_l^m(\theta, \phi),
  \end{eqnarray}
  where $R_l(r)$ solves an Euler type equation.  Thus
\begin{eqnarray}
\Lambda_\gamma Y_l^m(\theta,\phi)= \Lambda_\gamma u|_{r = 1} =  \left .\gamma \frac{\partial R_l }{\partial r}\right|_{r = 1} Y_l^m(\theta,\phi) =  \lambda_l Y_l^m(\theta,\phi). \label{eval1}
\end{eqnarray}
\end{proof}
Note that $\lambda$ is independent of $m$ since $R_l$ is independent
of $m$.

\subsection{Approximation of Eigenvalues and Eigenfunctions of the Dirichlet-to-Neumann Map}
\label{sec:eigvals}
Next we will consider how to approximate the eigenvalues
for the special case of a constant conductivity $\gamma=1$. The particular form of $R_l$ gives
the following result.
\begin{prop} The eigenvalues of $\Lambda_1$ are given by $\lambda_l = l$.
\end{prop}

In the case of a piecewise constant radially symmetric
conductivity the eigenvalues can be computed recursively. Suppose  $0 = r_0 <r_1 <
r_2<\ldots r_{N-1} < r_N = 1$ and  for $j = 1,2,\ldots,N$
\begin{eqnarray}\label{gammapw}
  \gamma (x) = \gamma_j >0,\qquad |x| \in [r_{j-1},r_j].
\end{eqnarray}

\begin{prop}
  Suppose $\gamma$ is given by \eqref{gammapw}. Then the eigenvalues of $\Lambda_\gamma$ are given by
  \begin{eqnarray*}
    \lambda_0 = 0,\quad \lambda_l = l - \frac{2l+1}{1+ C_{N-1}}, \quad l >0
  \end{eqnarray*}
where $C_j =  w_j\frac{\beta_l\gamma_{j+1} \rho_j + \gamma_j}{\gamma_{j+1}\rho_j - \gamma_j}$ with $\rho_1 = 1$, $\rho_j = \frac{C_{j-1}+ w_j}{C_{j-1} - \beta_l w_j}$, $\beta_l = \frac{l+1}{l}$ and $ w_j = r_j^{-(2l+1)}$.
\end{prop}
\begin{proof}  Since  $Y_0^0$ is a constant,  $\lambda_0 = 0$.
  The solution to $\nabla \cdot \gamma \nabla u_{lm} = 0, \; u_{lm}|_{\partial D} = Y_l^m,$ is given by \eqref{eq:sphericalsol} with $R_l(r) = A_j r^{l} + B_j
  r^{-(l+1)}$ for $r_{j-1} \leq r<r_j, \, j = 1, \ldots , N$. The
  coefficients $A_j$ and $B_j$ are determined by matching the
  Dirichlet and Neumann conditions at the $r_j$, $j=1,\ldots, N-1$. The outermost Dirichlet condition (at $r = 1$) gives
  $1 = A_N + B_N$ which leads to the following eigenvalue expression:
\begin{eqnarray}
\lambda_l = \gamma  \left .\frac {\partial v_{lm}}{\partial r}\right|_{\partial D} \quad
= lA_N -(l+1) B_N \quad
= l -(2l+1)B_N \label{eigval}
\end{eqnarray}

Moreover, by induction it follows that $A_j = B_jC_{j-1}$ for $j = 2,\ldots,N$.
Again using the Dirichlet condition from the boundary, $1= A_N + B_N$ we get $B_N= (C_{N-1} +1)^{-1}$ which leads to the expression of the eigenvalue as stated in the theorem.
\end{proof}

By \cite{somersaloCheneyIsaacsonIsaacson91} if conductivities $\gamma_L$ and $\gamma_U$ are such that $\gamma_L(r) \leq \gamma_U (r)$
for all $r$, then the eigenvalues
$\lambda_l^L$ and $\lambda_l^U$ of their corresponding Dirichlet-to-Neumann maps satisfy $\lambda_l^L \leq \lambda_l^U$. This gives a means for finding lower and upper bounds on the eigenvalues
of a smooth function by finding the eigenvalues of piecewise constant function, $\gamma_L$ and $\gamma_U$ that satisfy $\gamma_L(r)
\leq \gamma (r) \leq \gamma_U(r)$.

\section{Implementation details}\label{sec:numerics}

\subsection{Numerical method for computing the scattering transform $\t(\xi,\zeta)$}
We compute the scattering transform $\t(\xi,\zeta)$ from the
definition \eqref{t} as a comparison to the $\mathbf{t}^{\mbox{{\tiny exp}}}$ approximation and to study the reconstructions from an accurate scattering transform.  The computation requires that we solve the Lippmann-Schwinger equation \eqref{muLS} for $\mu(x,\zeta)$.    Hence we require
\begin{itemize}
\item A method of computation for the Faddeev Green's function in three dimensions
\item A numerical method for the solution of \eqref{muLS}
\item Numerical quadrature for computing $\t(\xi,\zeta)$ from \eqref{t}
\end{itemize}
 We describe each of these in turn.

\subsubsection{Computation of the Faddeev Green's function}

The Faddeev Green's function was defined in equations
\eqref{Gzeta}. The effect of scaling and rotation of $\zeta$ on
$G_\zeta$ was analyzed in \cite{corneanKnudsenSiltanen}, and it was
shown that when $\zeta$ satisfying $\zeta \cdot \zeta =0$ is
decomposed in the form
\begin{eqnarray}\label{zdecomp}
  \zeta = \kappa(k_\perp + i k),
\end{eqnarray}
where $k_\perp, k\in\mathbf R^3, |k_\perp| = |k|= 1, k \cdot k_\perp = 0,$
and $|\zeta| = \sqrt 2 \kappa,$
then
\begin{eqnarray}\label{sgzeta}
  g_\zeta(x) = \kappa^{n-2}g_{k_\perp + i k}(\kappa x).
\end{eqnarray}
Furthermore, if $R\in SO(2)$ then
\begin{eqnarray}\label{Rgzeta}
  g_\zeta(x) = g_{\mathbf{R}\zeta}(R x).
\end{eqnarray}
Combining \eqref{sgzeta} and \eqref{Rgzeta} yields the formula
\begin{eqnarray}
  g_\zeta(x) = \kappa g_{e_1+ie_2}(\kappa R x),
\end{eqnarray}
where $R\in SO(2)$ and the first and second column of $R$ is
$k,k_\perp$ respectively. This formula shows that it is sufficient to
compute $g_{e_1+ie_2}.$

To compute $g_{e_1+ie_2}$ we will use formula (6.4) of \cite{newton89}
\begin{eqnarray}\label{gzetaformula}
  g_{e_1+ie_2}(x) = \frac {e^{-r+x_2-ix_1}}{4\pi r} - \frac 1 {4\pi} \int_{s}^1 \frac {e^{-ru+x_2-ix_1}}{\sqrt{1-u^2}}J_1(r\sqrt{1-u^2})du,
\end{eqnarray}
where $J_1$ denotes the  Bessel function of the first find of order one.  Here $r = |x|$ and $s = \hat x \cdot e_2 =
x/|x|\cdot e_2.$ Since the function $J_1(t)/t$ is continuous on the
interval $[0,\infty)$ (in particular at $t = 0$), we will approximate the
integral in \eqref{gzetaformula} by a simple midpoint Riemann sum
\begin{eqnarray*}
  \int_{s}^1 \frac
  {e^{-ru+x_2-ix_1}}{\sqrt{1-u^2}}J_1(r\sqrt{1-u^2})du \approx
  \sum_{j=1}^N \frac
  {e^{-ru(j)+x_2-ix_1}}{\sqrt{1-u(j)^2}}J_1(r\sqrt{1-u(j)^2}) h,
\end{eqnarray*}
where $N$ is the number of discretization points, $h = (1-s)/N$ and
$u(j) = s + (j-1/2)h,\; j =1,2,\ldots, N.$

\subsubsection{The computation of complex geometrical optics}
Having computed the Faddeev Green's function we now turn to the
numerical solution of the integral equation \eqref{muLS} for
$\mu(\cdot, \zeta).$ We will use a method due to Vainikko  \cite{vainikko00} for solving
Lippmann-Schwinger equations; see also \cite{hohage,knudsenMuellerSiltanen04} for
implementations in different contexts.  The main idea is to transform \eqref{muLS} to a multiperiodic
integral equation in $\mathbf R^3,$ which can be solved efficiently using FFT.

Let $G_\rho =
\{x \in \mathbf R^3\; | \;
|x_i| \leq \rho \}.$ Then by assumption $\text{supp}(q) \subset \Omega \subset
G_1.$ Extend the potential $q$ and the Green's function $g_\zeta$
to $G_{2}$ such that
\begin{eqnarray*}
{q}^{\mbox{{\tiny p}}}(x) &= \left\{
\begin{array}{ll}
  q(x),  &x \in \Omega,\\
    0,  & x \in G_{2} \setminus \Omega,
\end{array}  \right.\qquad
  {g}^{\mbox{{\tiny p}}}_\zeta(x) &= \left\{
  \begin{array}{ll}
    g_\zeta(x),  &x \in \Omega,\\
    0,  & x \in G_{2} \setminus \Omega,
  \end{array}\right.
\end{eqnarray*}
and then extend  ${q}^{\mbox{{\tiny p}}}$ and ${g}^{\mbox{{\tiny p}}}_\zeta$ to $\mathbf R^3$ as
periodic functions in all variables with period equal to $4.$ Instead of \eqref{muLS} we consider the
periodic integral equation
\begin{eqnarray}\label{muLSperiodic}
  {\mu}^{\mbox{{\tiny p}}}(x,\zeta) + \int_{\mathbf R^3} {g}^{\mbox{{\tiny p}}}_\zeta(x-y)  {q}^{\mbox{{\tiny p}}}(y) {\mu}^{\mbox{{\tiny p}}}(y,\zeta) dy = 1.
\end{eqnarray}
This equation  is uniquely solvable  since  \eqref{muLS}  is, and
moreover one can show that on $\Omega$ we have
\begin{eqnarray*}
  {\mu}^{\mbox{{\tiny p}}}(x,\zeta) = \mu(x,\zeta),\qquad x\in\Omega.
\end{eqnarray*}
In order to solve \eqref{muLSperiodic} numerically define
\begin{eqnarray*}
  {\mathbb Z}_N^3 = \{ j \in {\mathbb Z}^3 \; | \; -N/2 \leq j_k < N/2,\; k = 1,\ldots,3 \}
\end{eqnarray*}
and the computational grid
\begin{eqnarray*}
  C_N = h {\mathbb Z}_N^3
\end{eqnarray*}
where $h =  4/N$ specifies the discretization fineness. Define the
grid approximation $\phi_N$ of a continuous function $\phi \in C(G_{2})$ by
\begin{eqnarray*}
  \phi_N(jh) = \phi(jh)
\end{eqnarray*}
and the grid approximation $g_N$ of ${g}^{\mbox{{\tiny p}}}_\zeta$ (which is smooth except for
a singularity at the origin) for fixed $\zeta$ by
\begin{eqnarray*}
  g_N(jh) = \left\{
  \begin{array}{ll}
    0, & j = 0\\
    {g}^{\mbox{{\tiny p}}}_\zeta(jh), & \textnormal {otherwise}.
  \end{array}\right.
\end{eqnarray*}
The convolution operator appearing in \eqref{muLSperiodic}
\begin{eqnarray*}
  K \phi (x) = \int_{\mathbf R^3} {g}^{\mbox{{\tiny p}}}_\zeta(x-y)  \phi(y) dy
\end{eqnarray*}
is now discretized by trigonometric collocation, which, using the
discrete Fourier transform ${\mathcal F}_N$, gives
\begin{eqnarray*}
   K_N \phi_N (jh) = {\mathcal F}_N^{-1} (\hat {g}^{\mbox{{\tiny p}}}_N \cdot \hat  \phi_N).
\end{eqnarray*}
Here $\cdot$ denotes pointwise multiplication. In practice the
discrete Fourier transform can be implemented efficiently using FFT (with
proper zero-padding) in $\O(N^3\log(N))$ arithmetic operations. The total
discretization of \eqref{muLSperiodic} now reads
\begin{eqnarray*}
    \mu_N + K_N(q_N\mu_N) = Q_N 1.
\end{eqnarray*}
This discrete linear system is solved numerically in matlab using the
iterative algorithm GMRES \cite{saadSchultz}, without setting up a
matrix for the linear map $K_N(q_N\cdot).$

\subsubsection{The scattering transform}
Having computed the grid approximation $\mu_N$ it is straightforward
to evalute $t(\xi,\zeta)$ by using numerical integration in \eqref{t}. In this
implementation we have used a simple midpoint quadrature rule.

\subsection{Numerical method for computing  $\mathbf{t}^{\mbox{{\tiny exp}}}$ for spherically symmetric conductivities}

For the calculation of $\mathbf{t}^{\mbox{{\tiny exp}}}$ we expand $e^{ix\cdot \zeta}$ in terms
of spherical harmonics\footnote{We use here the normalized spherical
  harmonics given by $Y_l^m(\theta,\phi) = N_l^mP_l^m(\cos \theta) e^{im\phi}$
  where $N_l^m$ are normalization factors and $P_l^m$ are associated
  Legendre functions.} and $e^{-ix \cdot(\xi+\zeta)}$ in terms of the
spherical harmonics conjugates,
\begin{eqnarray*}
 e^{-ix \cdot(\xi+\zeta)}&= \sum_{l=0}^\infty \sum_{m=-l}^{l} a^*_{lm}(\xi,\zeta) [Y_l^m(\theta,\phi)]^*\\
e^{ix\cdot \zeta} &= \sum_{k=0}^\infty \sum_{n=-k}^{k} b_{kn}(\zeta) Y_k^n(\theta,\phi).
\end{eqnarray*}
Using these expansions leads to
\begin{eqnarray}\label{eq:TexpSum}
\mathbf{t}^{\mbox{{\tiny exp}}}(\xi, \zeta) =\sum_{l,m} \sum_{k,n}  a^*_{lm}(\xi,\zeta) b_{kn}(\zeta)  \int_{\partial D} [Y_l^m(\theta,\phi)]^* (\Lambda_q -\Lambda_0)Y_k^n(\theta,\phi) d\sigma
\end{eqnarray}

In the special case of spherically symetric conductivities we can use the knowledge of the eigenvalues of the Dirichlet-to-Neumann maps, $\Lambda_\gamma Y_l^m(\theta,\phi) = \lambda_l Y_l^m$ to simplify the calculation of $\mathbf{t}^{\mbox{{\tiny exp}}}$. In particular we get
\begin{eqnarray}
\mathbf{t}^{\mbox{{\tiny exp}}}(\xi, \zeta)
&=&\sum_{l,m,k,n}  a^*_{lm}(\xi,\zeta)b_{kn}(\zeta)   \int_{\partial D} [Y_l^m(\theta,\phi)]^* (\lambda_k-k)Y_k^n(\theta,\phi) d\sigma\nonumber \\
&=&\sum_{l,m,k,n}  a^*_{lm}(\xi,\zeta)b_{kn}(\zeta)  (\lambda_k-k) \int_{\partial D} [Y_l^m(\theta,\phi)]^* Y_k^n(\theta,\phi) d\sigma\nonumber \\
&=&\sum_{l,m} a^*_{lm}(\xi,\zeta) b_{lm}(\zeta) (\lambda_l-l)
\label{texpassum}
\end{eqnarray}
The last equality comes from the orthonormality of the spherical harmonics. Equation (\ref{texpassum}) can be easily calculated if the coefficients $a^*_{lm}$ and $b_{lm}$  are available.
In this work, these coefficients were calculated with a  software package called `\textit{S2kit}' which are C routines that can be accessed from Matlab. Detailed information can be found in \cite{fastHarmExp03}.

\subsection{Computation of the conductivity}
After taking the high frequency limit in \eqref{eq:zetalim} and \eqref{texpzetalim}  we calculate the inverse
Fourier transform to get $q(x)$ and $q^{\mbox{{\tiny exp}}}(x).$ The integral in the inverse Fourier
transform is here computed numerically using a simple Riemann sum. To get the conductivity $\gamma$ we
need to solve the boundary value problem $\Delta \gamma^{1/2} =
q \gamma^{1/2} $ with $\gamma^{1/2}|_{\partial \Omega} = 1$. This was
realized with the standard Green's function for the Laplace equation
in three dimensions. Using symmetries reduces the problem to a single
integral.

\subsection{Numerical implementation of Calder\'on's method}
 Calder\'on's method based on \eqref{gammaapp} is
simply implemented by evaluating the integral using numerical quadrature.

\section{Results} \label{sec:examples}
\subsection{Examples}
\label{sec:theexample}
The conductivity distributions we will use in the examples are smooth,
spherically symmetric and constant one near $\partial\Omega$. They are given
by
 \begin{eqnarray}
 \gamma(x) &= (\alpha \Psi(|x|) +1)^2 \nonumber \\
 \Psi(r) &= \left \{ \begin{array}{cl} e^{-\frac{r^2}{(r^2-d^2)^2}}
     \quad &\mbox{for } -d<r<d \\
 0 \quad & \mbox{otherwise} \end{array} \right.\label{Psi}
 \end{eqnarray}
 where $0<d\leq 1$ is a parameter determining the support of $\Psi.$
 The parameter $\alpha$ regulates the amplitude of $\gamma$, which is
 largest at $r= 0$ with amplitude $(\alpha +1)^2$.  A similar function
 was used \cite{SMI00} as an example for the two dimensional problem.

 \subsection{The scattering transform} Let us fix $d = 0.9,\alpha =
 0.3.$ in \eqref{Psi}.  We are interested in the limit of ${\mathbf{t}}(\xi,\zeta), \xi
 \in \mathbf R^3,\zeta \in \V_\xi,$ when $|\zeta|$ goes to infinity. For
 purpose of illustration we compute ${\mathbf{t}}(\xi,\zeta)$ for fixed $\xi =
 (10,0,0)$ and varying $\zeta \in\V_\xi$ with $8<|\zeta| <50.$ We use
 a discretization level in the algorithm corresponding to $N=2^6.$ In
 addition we compute $\mathbf{t}^{\mbox{{\tiny exp}}}(\xi,\zeta)$ by \eqref{eq:TexpSum} using the first 30 eigenvalues
 of the Dirichlet-to-Neumann map.  We truncate the sum of the
 spherical harmonics at $l=30,$ which means we use approximately the
 first 900 spherical harmonics. As a benchmark we compute
 $\hat q(\xi).$ The results are shown in figure
 \ref{diffMagOfZeta}.
\begin{figure}[h!]
\begin{center}
\epsffile{figure_1.eps}
\caption[Scattering data ${\mathbf{t}}(\xi,\zeta)$ with varying $|\zeta|$]{$\t(\xi,\zeta),$
  $\mathbf{t}^{\mbox{{\tiny exp}}}(\xi,\zeta)$ calculated for fixed $\xi = (10, 0,0)$ and
  varying $|\zeta|$. Here $d=0.9$ and $\alpha=0.3$.}
\label{diffMagOfZeta}
\end{center}
\end{figure}
We know from Corollary~\ref{Treal} that $\t$ and $\mathbf{t}^{\mbox{{\tiny exp}}}$ are real and
this is consistent with our numerical results.  The data verifies that
for our example ${\mathbf{t}}(\xi,\zeta)$ converges to $\hat q (\xi)$ as $\zeta
\rightarrow \infty.$ We observe that $\mathbf{t}^{\mbox{{\tiny exp}}}$ is independent of the
magnitude of $\zeta \in \V_\xi,$ until it diverges due to numerical instability. The same phenomena appears in other examples and with
different values of $\xi.$ We believe that this phenomena has to do
with the special class of spherically symmetric conductivities considered
here.

Next we compare ${\mathbf{t}}(\xi,\zeta)$ and $\mathbf{t}^{\mbox{{\tiny exp}}}(\xi,\zeta)$ for different values of
$\xi$.  For each $\xi = s[1,0,0],\; s \in [0, 50],$ we fix $\zeta\in
\V_\xi$ with $|\zeta| = 50.$ We compute ${\mathbf{t}}(\xi,\zeta)$ using a discretization
level with $N = 2^6.$ $\mathbf{t}^{\mbox{{\tiny exp}}}$ is computed with the parameters as
above.   As a benchmark we compute $\hat q(\xi).$ The results are displayed in
figure \ref{qhat}.
\begin{figure}[h!]
\begin{center}
{\epsffile{figure_2.eps}}
\caption[Scattering data]{Scattering data $\t,$ $\mathbf{t}^{\mbox{{\tiny exp}}}$ and $\hat q$
  (qhat) with $d=0.9$ and $\alpha=0.3$. For each $\xi,$ $\zeta\in \V_\xi$ is
  chosen such $|\zeta| = 50.$ The Fourier transform $\hat q$ virtually coincides with
  $\t(\xi,\zeta)$. }
\label{qhat}
\end{center}
\end{figure}
The difference in $\hat q(\xi)$ and $\t(\xi,\zeta)$ is very small.
$\mathbf{t}^{\mbox{{\tiny exp}}}(\xi)$ is displayed only for $0\leq |\xi|\leq 32$ since the
calculation becomes numerically unstable and blows up for $|\xi|> 32$.
One observes good agreement of all three curves for $|\xi|\geq
20$. Close to $|\xi| = 0$ the approximation $\mathbf{t}^{\mbox{{\tiny exp}}}$ is close to zero
and differs from the correct values.

\subsection {The reconstructions}

Evaluting the inverse Fourier transform of the numerically computed $\mathbf{t}^{\mbox{{\tiny exp}}}(\xi)$ and
$\t(\xi,\zeta)$ gives two approximations of $q(x)$ which are displayed in figure \ref{q}.  The
approximation calculated from ${\mathbf{t}}(\xi,\zeta)$
differs as expected only slightly from the actual value. The
approximation $q^{\mbox{{\tiny exp}}}$ of $q$ calculated from $\mathbf{t}^{\mbox{{\tiny exp}}}$ (and hence from
the boundary data) is quite different from $q.$ For $x$ near the boundary the $q^{\mbox{{\tiny exp}}}(x)$ is quite accurate,
but for $x$ near zero there are large discrepancies, especially in the
magnitude.  Looking at the scattering data in figure \ref{qhat}, one
sees two features most likely responsible for that difference.  The
first one is the differences in the values of $\mathbf{t}^{\mbox{{\tiny exp}}}(\xi)$ for $\xi$
close to zero compared to $\hat q(\xi)$.  The second is the truncation
of $\mathbf{t}^{\mbox{{\tiny exp}}}(\xi)$ due to numerical instability for large $\xi$ values.
More details on the influence of the truncation of the scattering data
are provided in section \ref{sec:trunc}.
\begin{figure}[h!]
\begin{center}
{\epsffile{figure_3_1.eps}} \hfill {\epsffile{figure_3_2.eps}}
\caption[Reconstructions of the Schr\"odinger potential]{Left:  Reconstructions of $q(x)$ by taking the inverse Fourier transform of $\t(\xi,\zeta)$ and  $\mathbf{t}^{\mbox{{\tiny exp}}}(\xi,\zeta)$  for $\alpha = 0.3$ and $d =0.9.$  Right:   Reconstructions of $\gamma$ from $\t,$ $\mathbf{\gamma}^{\mbox{{\tiny exp}}},$ and $\mathbf{\gamma}^{\mbox{{\tiny app}}}$ compared to actual
  conductivity  for $\alpha = 0.3$ and $d = 0.9.$ $\gamma$ from $\t$ nearly coincides  with the actual conductivity.}
\label{q}
\end{center}
\end{figure}

Also in figure \ref{q} we display three reconstructions of the
conductivity distribution. The first reconstruction of $\gamma$ is from
${\mathbf{t}}(\xi,\zeta)$. Since ${\mathbf{t}}(\xi,\zeta)$ is computed from the forward problem, it may be expected that this reconstruction would be very close to the actual value, as it is. The second
reconstruction is $\mathbf{\gamma}^{\mbox{{\tiny exp}}}(x)$ from $\mathbf{t}^{\mbox{{\tiny exp}}},$ and the third
reconstruction $\mathbf{\gamma}^{\mbox{{\tiny app}}}$ is from the linear method
\eqref{gammaapp}. Considering the relatively large difference in
magnitude of $q^{\mbox{{\tiny exp}}}(x)$, the reconstruction $\mathbf{\gamma}^{\mbox{{\tiny exp}}}$ is
surprisingly good. Also $\mathbf{\gamma}^{\mbox{{\tiny app}}}$ is a fairly good reconstruction. A
positive aspect in both reconstructions is that we get $\gamma \equiv 1$ close to the
boundary. Moreover, the overall shape is also fairly well
reconstructed.

\subsection {The influence of the truncation of the scattering data}
\label{sec:trunc}

When we reconstructed $q^{\mbox{{\tiny exp}}}$ and $\mathbf{\gamma}^{\mbox{{\tiny exp}}}$ we truncated the scattering data $\mathbf{t}^{\mbox{{\tiny exp}}}$ due to numerical instabilities. In this section we investigate the
influence on the reconstructions of the truncation of the true scattering data ${\mathbf{t}}(\xi,\zeta).$
\begin{figure}[h!]
 {\epsffile{figure_4_1.eps}} \hfill {\epsffile{figure_4_2.eps}}
\caption[]{Left: reconstructed
   Schr\"odinger potential with truncation of $\t$ at $R=15, 25,$
   and $50.$ Right: reconstructions of ${\gamma}$.}
\label{fig:difftrunc}
\end{figure}
Figure \ref{fig:difftrunc} shows $\t(\xi,\zeta)$ and the reconstructions
$q(x)$ and ${\gamma(x)}$ for different truncations of $\t(\xi,\zeta)$,
namely at $\xi = R$ for $R = 15, 25, 50.$ We have chosen $\zeta \in
\V_\xi$ with $|\zeta| = 50.$ The actual potential and conductivity are
almost identical to the curves corresponding to $R=50.$  It is evident that the amount of truncation
of the scattering transform  influences the reconstruction, and
that a very poorly reconstructed $q$ can still result in a good
approximation of $\gamma$.

This suggests that for the reconstruction of $\gamma$ the values of the scattering data for small $\xi$ are very important.
This is analogous to observations made in the 2-D case \cite{MS03}.

\subsection {Influence of the support and magnitude of $\gamma(x)^{1/2}-1$}
So far we have used fixed values for $d$ and $\alpha$, which determine
the support and the magnitude of $\gamma(x)^{1/2}-1$.  Figure
\ref{fig:gamexpdiffddiffa} displays the reconstructions $\mathbf{\gamma}^{\mbox{{\tiny exp}}}$
and $\mathbf{\gamma}^{\mbox{{\tiny app}}}$ of
$\gamma(x)$ from $\mathbf{t}^{\mbox{{\tiny exp}}}(\xi)$ for different choices of support $d$ and
magnitude $\alpha.$  Each row corresponds to a certain $d$-value
and each column to a specific $\alpha$-value.  For small support and
small magnitude we get good reconstructions, but the quality changes dramatically with
larger amplitude and larger support. Especially $\mathbf{\gamma}^{\mbox{{\tiny exp}}}$ does not
recover the actual conductivity very well for the large amplitude
$\alpha = 0.9.$
\begin{figure}[h!]
\begin{center}
{\epsffile{figure_5.eps}}
\caption[Reconstructions with different support and magnitude of
$\gamma$]{Reconstructions of conductivities of varying support and magnitude: each row corresponds to a specific support $d$
and each column corresponds to a specific magnitude of $\gamma$. The dash-dotted
curves are the $\mathbf{\gamma}^{\mbox{{\tiny exp}}}$ reconstructions, the dashed curves are the
$\mathbf{\gamma}^{\mbox{{\tiny app}}}$ reconstructions, and solid curves are the
actual conductivities $\gamma.$}
\label{fig:gamexpdiffddiffa}
\end{center}
\end{figure}

\section{Conclusions}
In this work a direct method based on \cite{nachman88} for
reconstructing a 3-D conductivity distribution from the
Dirichlet-to-Neumann map was implemented and tested on noise-free data. A linearizing approximation
to the scattering transform, denoted $\mathbf{t}^{\mbox{{\tiny exp}}}$ was studied and compared
to Calder\'on's reconstruction algorithm.  Reconstructions of
spherically symmetric conductivities in the unit sphere were computed
using the $\mathbf{t}^{\mbox{{\tiny exp}}}$ approximation, Calder\'on's method, and a scattering
transform computed from the definition requiring knowledge of the
actual Schr\"odinger potential.  The latter case served as a benchmark
to study the quality of reconstructions for which the actual
scattering transform is known.  It was shown that very accurate
reconstructions can be obtained from accurate knowledge of the
scattering transform.  It was found that in contrast to the 2-D case,
the $\mathbf{t}^{\mbox{{\tiny exp}}}$ approximation is inaccurate near the origin, and this
results in poor approximations to the magnitude of the conductivity.
However, the support of $\gamma-1$ and the boundary value $\gamma=1$
was well approximated by all three methods.  Truncating the computed
scattering transform in the computations was found to have a profound
effect on the reconstructed Schr\"odinger potential $q$, but the
affect on the reconstructed conductivity $\gamma$ was less dramatic.
In summary, it appears that the use of the full scattering transform
in this method is a promising approach for 3-D reconstructions, while
linearizations lead to significant inaccuracies in the reconstructed
amplitudes.

\section*{Acknowledgments}
The authors thank D. Isaacson and G. Boverman for helpful discussions on the spherical harmonics.
This material is based upon work supported by the National Science Foundation under Grant No. 0513509 (J. Mueller).

\section*{References}

\providecommand{\bysame}{\leavevmode\hbox to3em{\hrulefill}\thinspace}
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\providecommand{\MRhref}[2]{%
  \href{http://www.ams.org/mathscinet-getitem?mr=#1}{#2}
}
\providecommand{\href}[2]{#2}

redpajama_set_name: RedPajamaArXiv


So you want to be an artist but you don't know how to pay for a college education or even where to begin to select the right program for you? This series answers those questions and prepares you (or your student) for the application and audition process. Sessions are available for students and parents alike, to build an environment for success.
(606) 303-3325 for further questions.
So Your Kid Wants to be an Artist?
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text:
Bigfoot Researcher Wants to Use Blimp with Thermal Imaging System to Find Sasquatch
November 6th, 2012 by Shane McGlaun
I don't know whether or not Bigfoot really exists. What I do know is that we find new species no one has ever seen with remarkable regularity. I'm not ready to rule out the existence of Bigfoot, but I'm also not ready to go running around the woods at night hunting for a hairy, smelly, primate.
A researcher from Idaho State University, Jeffrey Meldrum, is set to use a scientific approach to trying to determine once and for all if Bigfoot is real. Meldrum, a professor of anatomy and anthropology at the university, is trying to raise $300,000 to build a remote-control blimp that will carry a thermal imaging system. The idea is the blimp will float above forests searching for evidence of Bigfoot.
If the dirigible finds evidence of the beast, the location would be sent to people on the ground who would follow-up and try to catch an in-person glimpse of the fabled Sasquatch. Either that or the blimp could route them to their demise at the hands of a hungry and angry primate.
[Yahoo News via The Register]
Bigfoot Air Freshener Doesn't Actually Smell Like Sasquatch
This Drone Has Legs and Can Walk
Mysterious Heritage Limited Edition Ford Bronco Rumors Surface

redpajama_set_name: RedPajamaCommonCrawl


text:
Q: From likelihood P(Data|H) to P(H|Data) If there are four possible hypotheses and I calculate the likelihood of the data given each of these hypotheses, can I calculate the probability of one of the hypotheses as
P(H1|Data) = P(Data|H1) / (P(Data|H1)+P(Data|H2)+P(Data|H3)+P(Data|H4))
Sorry if it's too basic question
Thanks!

A: You're close, but not quite there yet.
Using Bayes' Theorem we know that: $$P(H_{1}|Data) = \frac{P(Data|H_{1}) P(H_1)}{P(Data)}$$.
However we cannot state that: $$P(Data) \neq \sum_{i}^{4} P(Data|H_{i})$$, from definition we know that $P(Data) = \sum_{i}^{4}P(Data \cap H_{i})$, which is equivalent to saying that  $$P(Data) = \sum_{i}^{4}P(Data|H_{i})*P(H_{i})$$.
So in order to fix your solution you would only have to scale each of the terms in the denominator with the corresponding probability of $P(H_{i})$.
So to summarize using your original notation we can compute $P(H_{1}|Data) = \frac{P(Data|H_{1}) P(H_1)}{P(Data|H_{1})*P(H_{1}) + P(Data|H_{2})*P(H_{2}) + P(Data|H_{3})*P(H_{3}) + P(Data|H_{4})*P(H_{4})}$

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text:
Insider reporting obligations and options backdating
Compton, Ryan and Sandler, Daniel and NIcholls, Christopher and Tedds, Lindsay M. (2011): Insider reporting obligations and options backdating. Published in: Banking and Finance Law Review , Vol. 25, No. 3 (2011)
In April 2010, new rules governing the reporting of securities trades by insiders of reporting issuers came into effect in Canada. These new rules were embodied in National Instrument 55-104, and in contemporaneous harmonized changes to Ontario's Securities Act. Under the new regime, the deadline for filing insider reports has been shortened, from ten calendar days to five calendar days following a purchase or sale. When a draft of National Instrument 55-104 was first published for comment, the Canadian Securities Administrators linked this proposed timing change, among other things, to the practice of improper stock options backdating. The suggestion that insider reporting obligations might deter the practice of options backdating is intriguing. Insider reporting rules have historically been regarded primarily as a regulatory tool to detect or prevent the improper use of inside information by insiders of reporting issuers. For these requirements to perform an effective secondary role in combating improper options backdating, clear rules on the timing of reporting obligations and rigorous enforcement would be required. It is not clear that the administration and enforcement of current Canadian insider reporting rules, crafted with very different objectives in mind, provide an effective deterrent to improper options backdating. A review of the current rules and the mechanisms for their enforcement, together with a comparison with insider reporting regimes in other selected jurisdictions, reveals weaknesses in the Canadian approach and suggests ways in which the Canadian regime could be enhanced to deter or detect options backdating.
Insider Reporting, Options Backdating, Securities Regulation, Canada
J - Labor and Demographic Economics > J3 - Wages, Compensation, and Labor Costs > J38 - Public Policy
J - Labor and Demographic Economics > J3 - Wages, Compensation, and Labor Costs > J33 - Compensation Packages ; Payment Methods
Lindsay Tedds
Ryan Compton, Daniel Sandler & Lindsay M. Tedds, "Options Backdating: A Canadian Perspective" (2009) 47 Can. Bus. L.J. 329.
M.P. Narayanan, Cindy A. Schipani & H. Nejat Seyhun, "The Economic Impact of Backdating of Executive Stock Options" (2007) 105 Mich. L.J. at 1597
Matthew S. Chambers, "Last Ditch Options: An Assessment of Independent Director Liability and a Proposal For Congressional Action in Light of the Employee Stock Option Back-Dating Scandal" (2008) 42 Ga. L. Rev. 569.
Philip Anisman, "The Proposals for a Securities Market Law for Canada: Purpose and Process" (1981) 19 Osgoode Hall L.J. 329
Marvin Yontef, "Insider Trading" in Proposals for a Securities Market Law for Canada, Vol. 3 (Ottawa: Minister of Supply and Services Canada, 1979) 625
Paul L. Davies, Gower and Davies' Principles of Modern Company Law, 7th ed (London: Sweet & Maxwell, 2003)
Janet Gamer Feldman and Richard L. Teberg, "Beneficial Ownership Under Section 16 of the Securities Exchange Act of 1934" 17 Case W. Res. L. Rev. (1965-1966) 1054
Gennaro Bernile & Gregg A. Jarell, "The Impact of the Options Backdating Scandal on Shareholders" (2009) 47:1 Journal of Accounting & Economics 2
Brian Hall and Kevin Murphy, "The Trouble with Stock Options" (2003) 17:3 Journal of Economic Perspectives 49
Erik Lie, "On the Timing of CEO Stock Option Awards" (2005) 51 Management Science 802.
David Yermack, "Good Timing: CEO Stock Option Awards and Company News Announcements" (1997), 52 Journal of Finance 449
Randall A. Heron & Erik Lie, "Does Backdating Explain the Stock Price Pattern around Executive Stock Option Grants?" (2007) 83 J. Fin. Econ. 271
Randall A. Heron & Erik Lie, "What Fraction of Stock Option Grants to Top Executives Have Been Backdated or Manipulated?" (2009) 55 Management Science 513
Daniel Collins, Guojin Gong & Haidan Li, "The Effect of the Sarbanes-Oxley Act on the Timing Manipulation of CEO Stock Option Awards" (2005)
David I. Walker, "Unpacking Backdating: Economic Analysis and Observations on the Stock Option Scandal" (2007) 87 B.U.L. Rev. 561
M.P. Narayanan & H. Nejat Seyhun, "Effect of Sarbanes-Oxley Act on the Influencing of Executive Compensation" (2005), online: SSRN http://ssrn.com/abstract=852964
Jesse Fried, "Option Backdating and its Implications" (2008) 65 Wash. & Lee L. Rev. 853
Lucian Bebchuk & Jesse M. Fried, "Paying for Long Term Performance"(2010) 158 U. Pa. L.R. 1915 at 1937
John Shipman, "The Future of Backdating Equity Options in the Wake of SEC Executive Compensation Disclosure Rules" (2007) 85 N.C.L. Rev. 1194.
Rory Cray, "Why the Options Backdating Scandal Won't Hit the UK" (24 January 2008), online: Out-Law.com (Pinsent Masons) <http://www.out-law.com/page-8825>.
Wise Persons' Committee to Review the Structure of Securities Regulation in Canada, It's Time (Ottawa: Department of Finance, 2003)
Hon. Peter deC Cory and Marilyn L. Pilkington, "Critical Issues in Enforcement", Research Study prepared for the Task Force to Modernize Securities Legislation in Canada, Canada Steps Up (2006), Vol. 6, 165 at 191-192.
William J. McNally & Brian F. Smith, "The Effect of Transparency on Insider Trading Disclosure" (2010) 36 Can. Pub. Pol'y 345 [McNally & Smith]


text:
\section{Introduction}

More than 99\% of the particles produced in heavy-ion collisions have
a momentum lower than $2~{\rm GeV}/c$. Therefore the study of these
``soft'' particles and the physical processes involved in their production
are of major interest for the comprehension of the Quark-Gluon Plasma
(QGP) properties: they give access to the chemical composition of the system, its size, its
temperature, and its dynamics. However, because of the large ``jump''
in energy, the abundancy of hard
processes at LHC will somehow modify our perception of soft physics:
high-$p_T$ mechanisms, which have already left their fingerprints at
RHIC, will certainly influence the medium in many respects.
For this reason, it will be essential to measure all the observables
relevant for heavy-ion data also in small systems such as \mbox{p-p}, first to get a hadronic reference for heavy-ion data
interpretation, and second to understand the underlying pQCD processes
at these energies.

At LHC, ALICE is the experiment best suited to study soft phenomena
due to its low material budget, its low magnetic field
and its main tracking device (the TPC), which provides a unique capability
to identify many particles down to low momenta $\sim 100~{\rm
MeV}/c$~\cite{Alessandro:2006yt}.
It is actually the only experiment at LHC almost fully dedicated
to the study of the QGP, though having a substantial physics
program for \mbox{p-p} to achieve.
An overall description of this experiment can
be found in ref.~\cite{Carminati:2004fp}.
The following sections will give an overview of ALICE performance in
the soft sector, showing results obtained for the following topics:
event characterization, (strange-) particle production, flow, femtoscopy and
event-by-event analyses.

\section{\label{sec:eventcharact}Event characterization and particle production}

All the heavy-ion collisions will be sorted as a function
of the impact parameter between the two incident nuclei, used to describe
the event ``centrality''. This is of major importance to derive the
involved number of participant nucleons and binary collisions, which are
critical elements in this field.
In ALICE the centrality is measured by combining the information from zero
degree hadronic and electromagnetic calorimeters located near the beam
axis. The distributions of their respective signals
are described in refs.~\cite{Alessandro:2006yt,Ramello:2008zz}.
The correlation of these two signals provide the number of spectators
in the collision, from which one
deduces the number of participants and the impact parameter, with
estimated resolutions of 15 nucleons and $1~{\rm fm}$ respectively~\cite{Alessandro:2006yt}.

Inclusive charged-particle multiplicity and pseudorapidity (${\rm d}N_{ch}/{\rm d}\eta$)
distributions supply additional event information: they
relate to the initial energy density of the medium, and help in
testing particle production models and understanding limiting fragmentation
phenomena (see~\cite{Busza:2007ke}.)
In ALICE these measurements are performed using the detectors ${\rm ITS+TPC}$
(central) and FMD (forward region) that allow a coverage of $-3.4<\eta<5.1$~\cite{Riedler:2006jq,Christensen:2007yc}.
The multiplicity and ${\rm d}N_{ch}/{\rm d}\eta$ distributions will be
the very first measurements ALICE will be able to achieve. The first
run should take place in fall 2009 with proton beams, at $\sqrt{s}$
equal to 0.9 and 10~TeV.

The amount of identified particles in ALICE will be sufficiently high to
provide temperature and baryochemical potential event by event. This
will give the possibility to place every event on the nuclear matter
phase diagram ($T$, $\mu_B$), which was not possible at lower energy heavy-ion accelerators.
Experimentally, that will be done by comparing the relative abundancies
between the different particles species to the ones expected by the
statistical models at equilibrium, which in turn furnishes the
parameters $T$ and $\mu_B$ that best describe the data.
These models have shown a good agreement with the data, for
example at RHIC~\cite{Andronic:2005yp}, which suggests that
chemical equilibrium is reached.
An inclusive measurement of all the particle species produced from
LHC \mbox{Pb-Pb} collisions will give insight on the question of chemical
equilibrium at an energy where hard processes are dominant.
It will be interesting to compare them
with the non-equilibrium scenario considered in
ref~\cite{Rafelski:2008an} which involves substantial strangeness
oversaturation.
A comparison to small ``canonical'' systems (\mbox{p-p}) will also be very
important to detect volume effects in particle production.
The measurement of unflavoured and especially strange species will strongly constrain these models.

Fig.~\ref{fig:ALICEpid} shows the range of transverse
momentum that ALICE will cover with the expected statistics of the first year of \mbox{Pb-Pb} data
taking, corresponding to $10^7$ central events.
At mid-rapidity,
identification of charged particles is performed via their energy loss (ITS, TPC) and time
of flight (TOF) up to momenta $\sim 2~{\rm GeV}/c$ in a full azimuthal
coverage. The TRD will in addition separate electrons from pions.
At higher momentum the HMPID is used, extending the identification
range of kaons and protons up to 5~GeV$/c$ in a limited azimuthal coverage.
In addition, identification of particle decays is done via invariant mass analysis (for
resonances) plus topological methods for strange and charm secondary
vertices. These methods improve substantially the accessible $p_T$ range, up to more than 10~GeV$/c$.

\begin{figure}[h]
\begin{center}
\includegraphics[scale=0.7]{ALICEpid.eps}
\end{center}
\caption{\label{fig:ALICEpid}Transverse momentum range of identification of
mesons and baryons at mid-rapidity, expected for $10^7$ \mbox{Pb-Pb} central
events at the energy $\sqrt{s_{NN}}=5.5~{\rm TeV}$. (Note: these
ranges are sensitive to the models used in the simulations.)}
\end{figure}

\section{Expansion and hadronization}
As the fireball expands and cools
down, the quarks hadronize and the system freezes-out. What happens
during this expansion and hadronization phase can be inferred from the study of the
dynamical properties of the resulting hadrons.
In this respect, the study of elliptic flow brings interesting
information (see~\cite{Adams:2005dq}.)
The measurement of the reaction plane is one of the
major issues in elliptic flow analysis. While the so-called ``event
plane'' method is sensitive to ``non-flow'' effects, the
``cumulant'' method doesn't give access to the reaction plane~\cite{Poskanzer:1998yz,Borghini:2001vi}.
A new possibility, based on both Lee-Yang Zeroes and event-plane methods has been
developed for ALICE; it provides the $v_2$ factor with a precision of $\sim 1\%$ (statistic error only), getting rid
of non-flow effets and auto-correlations, and the reaction plane with
a resolution factor of about 0.8~\cite{Bilandzic:2008nx,Kolk:2009}.

A good knowledge of the elliptic flow can supply substantial
information on the hadronization mechanisms. RHIC data have shown a
very good scaling of the $v_2$ factor with respect to the number of hadron
constituent quarks (two or three) as a function of transverse mass.
The coalescence models have been shown to provide one of the most accurate
explanations of this behaviour, and that has been corroborated by
baryon-over-meson ratio vs $p_T$ measurements~\cite{Fries:2008hs}.
In this context, the identification of strange secondary vertices
($K^0_S$, $\Lambda$, $\Xi$ and $\Omega$) plays an important role:
because their decay length is of several centimeters, their charged
decay modes can be reconstructed with topological methods, in which
single particle PID is not mandatory, and thus have no limitation
other than statistics.
As a result, the accessible $p_T$ range should go from $0.5$ to
at least 8-10~GeV$/c$ in the first \mbox{Pb-Pb} run of ALICE (see fig.~\ref{fig:ALICEpid}.)
This allows one to test the coalescence models and
identify the $p_T$ region above which fragmentation dominates
coalescence in hadron production.
Light resonances ($\rho$, $K^*$, $\phi$...) will give complementary
information on coalescence thanks to their range of identification in
$p_T$, but will also help probe chiral symmetry restoration~\cite{Pisarski:1981mq}
and the time between chemical and kinetic freeze-outs~\cite{Torrieri:2001ue}. In that
respect, the study of resonant-to-non-resonant particle ratios as a
function of event multiplicity and $p_T$ will be interesting.
It is expected that resonances can be identified up to $15~{\rm
GeV}/c$ in $p_T$, and recent studies show that PID is not mandatory
either~\cite{Badala:2008ga}, be it in \mbox{p-p} or heavy-ion collisions.

The interferometry method applied to identical particles is also a
powerful tool to probe the space-time evolution of the emitting source in
heavy-ions collisions, which will help constrain the hydrodynamical models.
The source radii observed at RHIC differ significantly from the
model predictions, possibly due to an incomplete hydrodynamic evolution of the fireball.
That measurement will therefore be fundamental at LHC. ALICE offers
great opportunities to perform these femtoscopic analyses thanks to
its identification performance at low $p_T$ and the expected high
event multiplicities. In such conditions, source radii up to 15~fm should be measurable~\cite{Kuhn:2008zza}.

\section{Event-by-event physics}

Fluctations of thermodynamical quantities from event to event are fundamental
to study the QCD phase transition and the order of this transition.
As a matter of fact, thanks to the identified particle multiplicities expected
in ALICE, various observables relative to the soft sector
will be analysed on an event-by-event basis, as mentioned in section~\ref{sec:eventcharact}.
For example, conserved quantities such as the net electrical charge
may provide information on the degrees of freedom of the initial state
of the collision, and thus directly on the phase transition;
fluctuations in temperature (extracted from particle $p_T$ spectra) should indicate the universality (or not) of the freeze-out
temperature.

Other interesting information will be obtained from the event-by-event
analysis program of ALICE.
Recent simulations using the Hijing particle generator show that
the relative error on reconstructed $K/\pi$ and $p/\pi$ ratios, event
by event, should not exceed a few percent; that will allow one to detect unexpected fluctuations due to QGP effects.
In addition, the balance functions, whose width is directly related
to the correlation range between particles, can probe the
hadronization time. A new feature possible in ALICE will be the study
of the balance function as function of $p_T$ for different rapidity gaps~\cite{Christakoglou:2009sq}.

\section{Conclusions}

The ALICE experiment at LHC has a wide program dedicated to the soft
phenomena. Both in \mbox{p-p} and \mbox{Pb-Pb}, the event multiplicity and particle pseudorapidity distributions will
be the first physics measurements ALICE will achieve. With the data
collected in just a few hours of \mbox{Pb-Pb} beams, many questions related to the medium
equilibrium, its equation of state, its expansion, and phase transition will be
assessed. The expected large event multiplicities will give the possibility
to address many observables on an event-by-event basis and thus
allow a novel approach, hardly possible at lower-energy accelerators, to investigate the underlying physics.
Obviously, all the soft physics results will have to be combined with
results coming from hard probes (including heavy flavours and jets), in order to reach a broad vision of
all the phenomena involved at these energies.

\section*{References}


Tender Care Veterinary Hospital connected with Think Up! in 2010 seeking help to develop a website which complemented their current brand and allowed easy management and updates. Leveraging the power of WordPress, we completed and executed a design for this holistic pet clinic which exceeded their needs. We also built in an email to fax functionality which used web forms for an easy client RX refill form to be submitted online and then received as a fax in the office. This streamlined the prescription refill process and created a much needed workflow improvement for office staff.


Broadview Entertainment Arts University awards transfer credits based on the guidelines found in the transfer credit section of the Broadview University course catalog.
The employment rate is a compilation of all BEAU graduates placement rates for this program. Placement rates were calculated for, and the school is accredited by, the Accrediting Council for Independent Colleges and Schools (ACICS). The placement rate includes students completing between July 1, 2014 and June 30, 2015.
The salary range is a compilation of BEAU graduates between July 1, 2012 and June 30, 2013 who reported salaries for this program.


Guantanamo Bay, Cuba, Jan 2017.
This is a picture of a guard at the prison. At Guantanamo you are under military censorship and you are not allowed to use a picture clearly identifying prisoners and guards so pictures of the prisoners or guards are backs or silhouettes. Every day the military go through your pictures and delete the once that are not approved.


Treat your loved one, friend or colleague to an assortment of chocolate FlavaBars® and FlavaMix® drinking chocolate all in an Earth toned, reusable FlavaNaturals® burlap bag. The perfect gift for any occasion. Say you care with chocolate!


The View's Editorial Staff has curated this year's best dressed teachers! See the finalists in this year-end video.
Love the idea! Super chic! We're just confused….did we miss the voting process?
Thanks! Oversight! We meant to include Ms. Atkinson!


William T. Freeman is the Thomas and Gerd Perkins Professor of Electrical Engineering and Computer Science at MIT, and a member of the Computer Science and Artificial Intelligence Laboratory (CSAIL) there. He was the Associate Department Head from 2011 - 2014.
His current research interests include machine learning applied to computer vision, Bayesian models of visual perception, and computational photography. He received outstanding paper awards at computer vision or machine learning conferences in 1997, 2006, 2009 and 2012, and test-of-time awards for papers from 1990, 1995 and 2005. Previous research topics include steerable filters and pyramids, orientation histograms, the generic viewpoint assumption, color constancy, computer vision for computer games, and belief propagation in networks with loops.


A City of Wetaskiwin citizen satisfaction survey highlights the city's strengths and areas upon which to improve. The results of the September survey were presented to council at its Nov. 14 meeting.
"In terms of resident issues, our top issues in 2016 included attracting more business, stores, shopping, restaurants — very similar to 2015 — roads and infrastructure … 11 per cent were concerned about high taxes and property taxes, and we saw a significant decline in that as being a concern," said Tracey With, vice president of Banister Research and Consulting INC.
She added the city's budget was also a concern and concern over crime and criminal activity doubled from last year to eight per cent. Development and growth of the city saw a decrease in significance, down to six per cent from 13 per cent.
With informed council 400 telephone interviews were conducted, including a cell phone sampling and interviews conducted with the city's younger population.
City of Wetaskiwin rating categories: quality of life, quality of services provided, management of community affairs, and management of growth and development all remained constant with 2015 or saw an increase. Management of growth and development was determined by Banister to have seen a "significant" increase, from 23 per cent to 30 per cent.
City council ratings saw a drop in four of its five areas, and a significant decrease in council's response to public inquiries: 42 per cent last year and 34 per cent this year.
Voters gave a better standing to council's ability to reflect their views in council decisions, up to 32 per cent from 30 per cent.
"Overall ratings, Wetaskiwin continues to rate high," said With.
When it comes to quality of services and facilities there was a display of mixed results. "We did see a bit of a decrease in the RCMP this year," said With. RCMP was labeled as a significant decrease, down to 85 per cent from last year's 93 per cent.
"We did link this to terms of safety," she added.
With explained survey results indicate, within areas of public safety, overall consideration of the community as a safe place to live went down significantly, down to 64 per cent from 75 per cent. Feelings of safety walking alone in the dark decreased and feelings of safety walking alone in daylight stayed even with last year at 90 per cent.
Ratings of outdoor recreation facilities increased significantly, up 11 per cent from 69 per cent; recreation programming increased significantly, 69 per cent from 58 per cent; and the Wetaskiwin Municipal Airport increased significantly, up to 46 per cent from 39 per cent.
Perceived value from property taxes; 39 per cent of survey participants feel they are getting good value, up eight per cent from last year. The next highest bracket, at 25 per cent, feel the value for their money is neither good nor poor.
A majority no — 67 per cent — indicate they are not willing to pay more for improved services. This remains the same as in 2015.
In terms of services improvements, this year road, sidewalk and repair services decreased in importance from last year and 41 per cent of participants have no opinion of specific improvements mentioned.
"Your staff are busier this year, plain and simple," said With. Face-to-face communication increased and phone communication decreased.
"Customer service ratings did see a bit of a slide," said With.
At 68 per cent, customer service expectations are what people were expecting — the same as last year. Being treated worse than expected increased two per cent over last year to 11 per cent.
As main sources of information, the city's website remains the top source and is climbing from last year. The telephone book is the second top source. The Wetaskiwin Times newspaper declined two per cent as a source to get information and the Pipestone Flyer newspaper saw an increase of one per cent over last year.
Parade of Programs, What's Happening in Wetaskiwin and the city's website all increased significantly as effective means of communications. Frequency of city website use declined significantly from 47 per cent to 31 per cent. Familiarity of the website did see an increase of four per cent.
The presentation was accepted by city council as information.


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But I went to a small liberal arts college and had to take nonscience subjects, too.
Teaching biotechnology through case studies - Can we improve higher order thinking skills of nonscience majors?
Song P Bacterial monologue: an engaging writing activity for nonscience majors.
Science motivation questionnaire II: validation with science majors and nonscience majors.
It is important to have books like A Sugar Creek Chronicle written for nonscience citizens.
Science Motivation Questionnaire: Construct Validation with Nonscience Majors.
In addition, the presence of each of the following themes was coded: 1) human impact or human-environment relationship, 2) policy implications (including "sustainability" or "remediation"), 3) ethics or philosophy, 4) for nonscience majors, and 5) for teacher certification.
Abele, Penprase, and Ternes (2013) conducted a retrospective review of the records of nursing students from a Midwestern university to see if their performance in pre-nursing, nonscience courses could serve as predictors of success in nursing.
Course-based research experiences (CREs) are powerful strategies to spread these gains in learning and persistence to all students, both science majors and nonscience majors (AAAS 2011).
I want to address a different but related point, agreeing with the bill's sponsor, Lamar Smith, that it is incumbent on the publicly funded science community--just as it is on elected and appointed government officials--to say and convey to the nonscience public: "I work for you--and I look forward to telling you how, and to answering your questions.
His publications include Science, Nonscience, and Nonsense: Approaching Environmental Literacy (The Johns Hopkins University Press, 1995) and a blog for The Huffington Post.


text:
Nilkantha Uprety's Appointment In Controversy
A decision of the government to nominate former chief election commissioner Nilkantha Uprety as an ambassador to India has landed in constitutional controversy regarding whether the position of ambassador is political or administrative.
The cabinet meeting proposed former chief election commissioner as ambassador to India. Article 245 clause 8 says not the persons who serve as a head of constitutional body is not eligible for the government appointment. However, the article also says this does not mean that it bars appointment in political position. The debate now intensifies whether ambassador is a political or administrative position.
Constitutional lawyer Chandrakanta Gyawali said the position of ambassador is administrative position and the appointment is unconstitutional. Former secretary Bhim Upadyaya also sees the position of ambassador is an administrative. However, government defended the decision saying that the position is political in nature Dessanchar reports.